Algebra

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

      Re-Imagining Video-based Online Learning

      Despite the tremendous growth in the availability of mathematics videos online, little research has investigated student learning from them. The goal of this exploratory project is to create, investigate, and provide evidence of promise for a model of online videos that embodies a more expansive vision of both the nature of the content and the pedagogical approach than is currently represented in YouTube-style lessons.

      Award Number: 
      1416789
      Funding Period: 
      Mon, 09/01/2014 to Fri, 08/31/2018
      Full Description: 

      The goal of this exploratory project is to create, investigate, and provide evidence of promise for a model of online videos that embodies a more expansive vision of both the nature of the content and the pedagogical approach than is currently represented in YouTube-style lessons. This goal is pursued through the development and research of videos for two mathematics units--one focused on proportional reasoning at the middle grades level and the other focused on quadratic functions at the high school level, using an approach that could be applied to any STEM content area. The media attention on the Khan Academy and the wide array of massive open online courses has highlighted the internet phenomenon of widespread accessibility to mathematics lessons, which offer many benefits, such as student control of the pace of learning and earlier access to advanced topics than is often possible in public schools. Yet, despite the huge range of topics presented in online videos, there is surprising uniformity in the procedural emphasis of the content and in the expository mode of presentation. Moving beyond the types of videos now used, primarily recorded lectures that replicate traditional classroom experience, this project advances our understanding about how students learn from video and from watching others learn - vicarious learning - as opposed to watching an expert. This project addresses the need for an alternative approach. Rather than relying on an expository style, the videos produced for this project focus on pairs of students, highlighting their dialogue, explanations and alternative conceptions. This alternative has the potential to contribute to learning sciences and to develop a usable tool.

      Despite the tremendous growth in the availability of mathematics videos online, little research has investigated student learning from them. This project develops dialogue-intensive videos in which children justify and explain their reasoning, elucidate their own comprehension of mathematical situations, and argue for and against various ideas and strategies. According to Wegerif (2007), such vicarious participation in a dialogic community may help learners take the perspective of another in a discussion, thus "expanding the spaces of learning" through digital technology. Consequently, a major contribution of this proposed work will be a set of four vicarious learning studies. Two qualitative studies investigate the particular meanings and ways of reasoning that learners appropriate from observing the dialogue of the students in the videos, as well as the learning trajectories of vicarious learners for each unit. Two quantitative studies isolate and test the effectiveness of the dialogic and the conceptual components of the model by comparing learning outcome gains for (a) conceptual dialogic versus conceptual expository conditions, and (b) dialogic conceptual versus dialogic procedural conditions. Another mark of the originality of the proposed work is the set of vicarious learning studies that contributes to the emerging literature across several dimensions, by (a) using secondary students rather than undergraduates; (b) exploring longer periods of learning, which is more conducive to deeper understanding; and (c) examining the nature of reasoning that is possible, not just the effectiveness of the approach.

      Preparing Urban Middle Grades Mathematics Teachers to Teach Argumentation Throughout the School Year

      The objective of this project is to develop a toolkit of resources and practices that will help inservice middle grades mathematics teachers support mathematical argumentation throughout the school year. A coherent, portable, two-year-long professional development program on mathematical argumentation has the potential to increase access to mathematical argumentation for students nationwide and, in particular, to address the needs of teachers and students in urban areas.

      Lead Organization(s): 
      Award Number: 
      1417895
      Funding Period: 
      Sun, 06/15/2014 to Thu, 05/31/2018
      Full Description: 

      The project is an important study that builds on prior research to bring a comprehensive professional development program to another urban school district, The District of Columbia Public Schools. The objective of this full research and development project is to develop a toolkit  that provides resources and practices for inservice middle grades mathematics teachers to support mathematical argumentation throughout the school year. Mathematical argumentation, the construction and critique of mathematical conjectures and justifications, is a fundamental disciplinary practice in mathematics that students often never master. Building on a proof of concept of the project's approach ifrom two prior NSF-funded studies, this project expands the model to help teachers support mathematical argumentation all year. A coherent, portable, two-year-long professional development program on mathematical argumentation has the potential to increase access to mathematical argumentation for students nationwide and, in particular, to address the needs of teachers and students in urban areas. Demonstrating this program in the nation's capital will likely attract broad interest and produces important knowledge about how to implement mathematical practices in urban settings. Increasing mathematical argumentation in schools has the potential for dramatic contributions to students' achievement and participation in 21st century workplaces.

      Mathematical argumentation is rich discussion in which students take on mathematical authority and co-construct conjectures and justifications. For many teachers, supporting such discourse is challenging; many are most comfortable with Initiate-Respond-Evaluate types of practices and/or have insufficient content understanding. The professional development trains teachers to be disciplined improvisers -- professionals with a toolkit of tools, knowledge, and practices to be deployed creatively and responsively as mathematical argumentation unfolds. This discipline includes establishing classroom norms and planning lessons for argumentation. The model's theory of action has four design principles: provide the toolkit, use simulations of the classroom to practice improvising, support learning of key content, and provide job-embedded, technology-enabled supports for using new practices all year. Three yearlong studies will address design, feasibility, and promise. In Study 1 the team co-designs tools with District of Columbia Public Schools staff. Study 2 is a feasibility study to examine program implementation, identify barriers and facilitators, and inform improvements. Study 3 is a quasi-experimental pilot to test the promise for achieving intended outcomes: expanding teachers' content knowledge and support of mathematical argumentation, and increasing students' mathematical argumentation in the classroom and spoken argumentation proficiency. The studies will result in a yearlong professional development program with documentation of the theory of action, design decisions, pilot data, and instrument technical qualities.

      Learning Trajectories in Grades K-2 Children's Understanding of Algebraic Relationships

      This project will use classroom-based research to teach children about important algebraic concepts and to carefully explore how children come to understand these concepts. The primary goal is to identify levels of sophistication in children's thinking as it develops through instruction. Understanding how children's thinking develops will provide a critical foundation for designing curricula, developing content standards, and informing educational policies.

      Lead Organization(s): 
      Award Number: 
      1415509
      Funding Period: 
      Tue, 07/15/2014 to Thu, 06/30/2016
      Full Description: 

      Algebra is a central concern in school mathematics education. Its historical gatekeeper role in limiting students' career and life choices is well documented. In recent years, the response has been to reframe algebra as a K-12 endeavor. To this end, research on children's algebraic thinking in grades 3-5 shows that students can begin to understand algebraic concepts in elementary grades that they will later explore more formally. However, there is much that is unknown about how children in grades K-2 make sense of algebraic concepts appropriate for their age. This project aims to understand specific ways in which grades K-2 children begin to think algebraically. It will identify how children understand mathematical relationships, how they represent the relationships they notice, and how they use these relationships as building blocks for more sophisticated thinking. The project will use classroom-based research to teach children about important algebraic concepts and to carefully explore how children come to understand these concepts. The primary goal is to identify levels of sophistication in children's thinking as it develops through instruction. Understanding how children's thinking develops will provide a critical foundation for designing curricula, developing content standards, and informing educational policies, all in ways that can help children become successful in algebra and have wider access to STEM-related careers.

      While college and career readiness standards point to the role of algebra beginning in kindergarten, the limited research base in grades K-2 restricts algebra's potential in K-2 classrooms. This project will develop cognitive foundations regarding how children learn to generalize, represent, and reason with algebraic relationships. Such findings will inform both the design of new interventions and resources to strengthen algebra learning in grades K-2 and the improvement of educational policies, practices, and resources. The project will use design research to identify: (1) learning trajectories as cognitive models of how grades K-2 children learn to generalize, represent, and reason with algebraic relationships within content dimensions where these practices can occur (e.g., generalized arithmetic); (2) critical junctures in the development of these trajectories; and (3) characteristics of tasks and instruction that facilitate movement along the trajectories. The project's design will include the use of classroom teaching experiments that incorporate: (1) instructional design and planning; (2) ongoing analysis of classroom events; and (3) retrospective analysis of all data sources generated in the course of the experiment. This will allow for the development and empirical validation of hypothesized trajectories in students' understanding of algebraic relationships. This exploratory research will contribute critical early-grade cognitive foundations of K-12 teaching and learning algebra that can help democratize access to student populations historically marginalized by a traditional approach to teaching algebra. Moreover, the project will occur in demographically diverse school districts, thereby increasing the generalizability of findings across settings.

      Intensified Algebra Efficacy Study

      Students who fail algebra in the ninth grade are significantly less likely than their peers to graduate from high school on time. This project intends to test a common support strategy for at-risk students that provides an extra period of algebra, commonly known as a "double dose" condition. The Intensified Algebra (IA) is an intervention that addresses both the academic and non-academic needs of students.

      Partner Organization(s): 
      Award Number: 
      1418267
      Funding Period: 
      Fri, 08/01/2014 to Tue, 07/31/2018
      Full Description: 

      Student success in algebra continues to be a problem as many U.S. students are underprepared when they enter high school. Students who fail algebra in the ninth grade are significantly less likely than their peers to graduate from high school on time. This project intends to test a common support strategy for at-risk students that provides an extra period of algebra, commonly known as a "double dose" condition. The Intensified Algebra (IA) is an intervention that addresses both the academic and non-academic needs of students. It is set of cohesive, integrated, and rigorous resources that builds student motivation and confidence. IA uses a blended model of instruction with a strong technology component designed to support the productive use of expanded instructional time that has shown evidence of promise in earlier studies.

      This project is intended to rigorously test the impact of IA on student outcomes in a school-level random assignment design involving 6 districts, 55 high schools and over 4,000 9th grade students across two cohorts. Within each district, eligible schools are randomly assigned to either implement IA or to use the school's already established "double dose" algebra course. Analyses will use hierarchical linear models that explicitly take into account the clustering of students within classrooms and classrooms within schools. The study investigates short-term outcomes including end-of-9th grade algebra learning, passing rate for algebra I and attitudes toward mathematics. Longer-term outcomes include subsequent course-taking patterns and performances. The study examines fidelity of implementation and key implementation factors with descriptive and correlational analyses.

      GRIDS: Graphing Research on Inquiry with Data in Science

      The Graphing Research on Inquiry with Data in Science (GRIDS) project will investigate strategies to improve middle school students' science learning by focusing on student ability to interpret and use graphs. GRIDS will undertake a comprehensive program to address the need for improved graph comprehension. The project will create, study, and disseminate technology-based assessments, technologies that aid graph interpretation, instructional designs, professional development, and learning materials.

      Award Number: 
      1418423
      Funding Period: 
      Mon, 09/01/2014 to Sat, 08/31/2019
      Full Description: 

      The Graphing Research on Inquiry with Data in Science (GRIDS) project is a four-year full design and development proposal, addressing the learning strand, submitted to the DR K-12 program at the NSF. GRIDS will investigate strategies to improve middle school students' science learning by focusing on student ability to interpret and use graphs. In middle school math, students typically graph only linear functions and rarely encounter features used in science, such as units, scientific notation, non-integer values, noise, cycles, and exponentials. Science teachers rarely teach about the graph features needed in science, so students are left to learn science without recourse to what is inarguably a key tool in learning and doing science. GRIDS will undertake a comprehensive program to address the need for improved graph comprehension. The project will create, study, and disseminate technology-based assessments, technologies that aid graph interpretation, instructional designs, professional development, and learning materials.

      GRIDS will start by developing the GRIDS Graphing Inventory (GGI), an online, research-based measure of graphing skills that are relevant to middle school science. The project will address gaps revealed by the GGI by designing instructional activities that feature powerful digital technologies including automated guidance based on analysis of student generated graphs and student writing about graphs. These materials will be tested in classroom comparison studies using the GGI to assess both annual and longitudinal progress. Approximately 30 teachers selected from 10 public middle schools will participate in the project, along with approximately 4,000 students in their classrooms. A series of design studies will be conducted to create and test ten units of study and associated assessments, and a minimum of 30 comparison studies will be conducted to optimize instructional strategies. The comparison studies will include a minimum of 5 experiments per term, each with 6 teachers and their 600-800 students. The project will develop supports for teachers to guide students to use graphs and science knowledge to deepen understanding, and to develop agency and identity as science learners.

      Changing Culture in Robotics Classroom (Collaborative Research: Shoop)

      Computational and algorithmic thinking are new basic skills for the 21st century. Unfortunately few K-12 schools in the United States offer significant courses that address learning these skills. However many schools do offer robotics courses. These courses can incorporate computational thinking instruction but frequently do not. This research project aims to address this problem by developing a comprehensive set of resources designed to address teacher preparation, course content, and access to resources.

      Lead Organization(s): 
      Partner Organization(s): 
      Award Number: 
      1418199
      Funding Period: 
      Mon, 09/01/2014 to Thu, 08/31/2017
      Full Description: 

      Computational and algorithmic thinking are new basic skills for the 21st century. Unfortunately few K-12 schools in the United States offer significant courses that address learning these skills. However many schools do offer robotics courses. These courses can incorporate computational thinking instruction but frequently do not. This research project aims to address this problem by developing a comprehensive set of resources designed to address teacher preparation, course content, and access to resources. This project builds upon a ten year collaboration between Carnegie Mellon's Robotics Academy and the University of Pittsburgh's Learning Research and Development Center that studied how teachers implement robotics education in their classrooms and developed curricula that led to significant learning gains. This project will address the following three questions:

      1.What kinds of resources are useful for motivating and preparing teachers to teach computational thinking and for students to learn computational thinking?

      2.Where do teachers struggle most in teaching computational thinking principles and what kinds of supports are needed to address these weaknesses?

      3.Can virtual environments be used to significantly increase access to computational thinking principles?

      The project will augment traditional robotics classrooms and competitions with Robot Virtual World (RVW) that will scaffold student access to higher-order problems. These virtual robots look just like real-world robots and will be programmed using identical tools but have zero mechanical error. Because dealing with sensor, mechanical, and actuator error adds significant noise to the feedback students' receive when programming traditional robots (thus decreasing the learning of computational principles), the use of virtual robots will increase the learning of robot planning tasks which increases learning of computational thinking principles. The use of RVW will allow the development of new Model-Eliciting Activities using new virtual robotics challenges that reward creativity, abstraction, algorithms, and higher level programming concepts to solve them. New curriculum will be developed for the advanced concepts to be incorporated into existing curriculum materials. The curriculum and learning strategies will be implemented in the classroom following teacher professional development focusing on computational thinking principles. The opportunities for incorporating computationally thinking principles in the RVW challenges will be assessed using detailed task analyses. Additionally regression analyses of log-files will be done to determine where students have difficulties. Observations of classrooms, surveys of students and teachers, and think-alouds will be used to assess the effectiveness of the curricula in addition to pre-and post- tests to determine student learning outcomes.

      Illuminating Learning by Splitting: A Learning Analytics Approach to Fraction Game Data Analysis

      This project uses learning analytics and educational data mining methods to examine how elementary students learn in an online game designed to teach fractions using the splitting model. The project uses data to examine the following questions: 1) Is splitting an effective way to learn fractions?; 2) How do students learn by splitting?; 3) Are there common pathways students follow as they learn by splitting?; and 4) Are there optimal pathways for diverse learners?

      Lead Organization(s): 
      Partner Organization(s): 
      Award Number: 
      1338176
      Funding Period: 
      Mon, 07/01/2013 to Wed, 12/31/2014
      Full Description: 

      Mathematical literacy is a critical need in our increasingly technological society. Fractions have been identified as a key area of understanding, both for success in Algebra and for access to higher-level mathematics. The project uses learning analytics and educational data mining methods to examine how elementary students learn in Refraction, an online game designed to teach fractions using the splitting model. The project uses the data from a pre- and posttest of fraction understanding and log data from 3000 third-grade students' gameplay to examine the following questions:

      1) Is splitting an effective way to learn fractions?

      2) How do students learn by splitting?

      3) Are there common pathways students follow as they learn by splitting?

      4) Are there optimal pathways for diverse learners?

      Splitting is a well-known theory of fraction learning and has significant expert buy in. However, few of the research questions above can be advanced past the field's present level of understanding with either current qualitative or quantitative methods. By using data mining methods such as cluster analysis, association rule mining, and predictive analysis, the project provides numerous insights about student learning through splitting, including: classification of learning profiles exhibited in unstructured learning environments, common mistakes and sense-making patterns, the value or cost of exploration in learning, and the best path through learning for different students (such as those who score low on a pre-test).

      The project staff shares the methods and results through traditional and novel outlets for maximum impact on the field and on policy. In addition to conferences and journal publications, the principal investigator is working in several contexts in which this work is an exemplar of new ways the field can develop understanding of learning. In addition, many of these contexts have connections to efforts such as the Chief State School Officers' Shared Learning Collaborative, leading to a high probability that the findings and products can quickly impact large numbers of schools across the country.

      Using Math Pathways and Pitfalls to Promote Algebra Readiness

      This project that creates a set of materials for middle grades students and teacher professional development that would support the learning of early algebra. Building on their prior work with an elementary version, the efficacy study focuses on the implementation of the principals underlying the materials, fidelity of use of the materials, and impact on students' learning.

      Lead Organization(s): 
      Award Number: 
      1314416
      Funding Period: 
      Tue, 10/01/2013 to Sat, 09/30/2017
      Full Description: 

      Using Math Pathways & Pitfalls to Promote Algebra Readiness is a 4-year Full Research and Development project that creates a set of materials for middle grades students and teacher professional development that would support the learning of early algebra. Building on their prior work with an elementary version, the efficacy study focuses on the implementation of the principals underlying the materials, fidelity of use of the materials, and impact on students' learning.

      The project's goals are to: 1) develop an MPP book and companion materials dedicated to algebra readiness content and skills, 2) investigate how MPP transforms pedagogical practices to improve students' algebra readiness and metacognitive skills, and 3) validate MPP's effectiveness for improving students' algebra readiness with a large-scale randomized controlled trial.

      The iterative design and efficacy studies produce research-based materials to increase student learning of core concepts in algebra readiness. Though the focus of the project is algebra readiness, the study also examines the validity of the pedagogical approach of MPP. The MPP lesson structures are designed to help students confront common misconceptions, dubbed "pitfalls," through sense-making, class discussions, and the use of multiple visual representations. If the pedagogical approach of MPP proves to be successful, the lesson structures can be presented as an effective framework for instruction that extends to other content areas in mathematics and other disciplines.

      The project addresses a critical need in education, and the potential impact is large. Math achievement in the U.S. is not keeping pace with international performance. The current project focuses on algebra readiness skills, an area that is critical for future success in mathematics. Algebra often serves as a gatekeeper to more advanced mathematics, and performance in algebra has been linked to success in college and long-term earnings potential. Longitudinal studies indicate that students taking rigorous high school mathematics courses are twice as likely to graduate from college as those who do not. Thus, adequately preparing students for algebra can dramatically affect educational outcomes for students. The current project broadens the participation of underrepresented groups of students in math and later science classes that require strong math skills. The intervention builds on materials and pedagogical techniques that have demonstrated positive outcomes for diverse students. The targeted districts have large samples of English language learners and students from groups traditionally underrepresented in STEM so that we may evaluate the impact of the intervention on these populations. At the end of the project, the publication quality materials will be readily available to teachers and districts through our website www.wested.org/mpp.

      Learning Algebra and Methods for Proving (LAMP)

      This project tests and refines a hypothetical learning trajectory and corresponding assessments, based on the collective work of 50 years of research in mathematics education and psychology, for improving students' ability to reason, prove, and argue mathematically in the context of algebra. The study produces an evidence-based learning trajectory and appropriate instruments for assessing it.

      Lead Organization(s): 
      Partner Organization(s): 
      Award Number: 
      1317034
      Funding Period: 
      Tue, 10/01/2013 to Wed, 09/30/2015
      Full Description: 

      The Learning Algebra and Methods for Proving (LAMP) project tests and refines a hypothetical learning trajectory and corresponding assessments, based on the collective work of 50 years of research in mathematics education and psychology, for improving students' ability to reason, prove, and argue mathematically in the context of algebra. The goals of LAMP are: 1) to produce a set of evidence-based curriculum materials for improving student learning of reasoning, proving, and argumentation in eighth-grade classrooms where algebra is taught; 2) to produce empirical evidence that forms the basis for scaling the project to a full research and development project; and 3) to refine a set of instruments and data collection methods to support a full research and development project. LAMP combines qualitative and quantitative methods to refine and test a hypothetical learning trajectory for learning methods of reasoning, argumentation, and proof in the context of eighth-grade algebra curricula. Using qualitative methods and quantitative methods, the project conducts a pilot study that can be scaled up in future studies. The study produces an evidence-based learning trajectory and appropriate instruments for assessing it.

      Over the past two decades, national organizations have called for more attention to the topics of proof, proving, and argumentation at all grade levels. However, the teaching of reasoning and proving remains sparse in classrooms at all levels. LAMP will address this critical need in STEM education by demonstrating ways to improve students' reasoning and argumentation skills to meet the demands of college and career readiness.

      This project promises to have broad impacts on future curricula in the United States by creating a detailed description of how to facilitate reasoning and argumentation learning in actual eighth-grade classrooms. At present, a comprehensive understanding of how reasoning and proving skills develop alongside algebraic thinking does not exist. Traditional, entirely formal approaches such as two-column proof have not demonstrated effectiveness in learning about proof and proving, nor in improving other mathematical practices such as problem-solving skills and sense making. While several studies, including studies in the psychology literature, lay the foundation for developing particular understandings, knowledge, and skills needed for writing viable arguments and critiquing the arguments of others, a coherent and complete set of materials that brings all of these foundations together does not exist. The project will test the hypothetical learning trajectory with classrooms with high proportions of Native American students.

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