Grounded Theory

CAREER: Investigating Changes in Students' Prior Mathematical Reasoning: An Exploration of Backward Transfer Effects in School Algebra

This project explores "backward transfer", or the ways in which new learning impacts previously-established ways of reasoning. The PI will observe and evaluate algebra I students as they learn quadratic functions and examine how different kinds of instruction about the new concept of quadratic functions helps or hinders students' prior mathematical knowledge of the previous concept of linear functions.

Lead Organization(s): 
Award Number: 
1651571
Funding Period: 
Sat, 07/01/2017 to Thu, 06/30/2022
Full Description: 

As students learn new mathematical concepts, teachers need to ensure that prior knowledge and prior ways understanding are not negatively affected. This award explores "backward transfer", or the ways in which new learning impacts previously-established ways of reasoning. The PI will observe and evaluate students in four Algebra I classrooms as they learn quadratic functions. The PI will examine how different kinds of instruction about the new concept of quadratic functions helps or hinders students' prior mathematical knowledge of the previous concept of linear functions. More generally, this award will contribute to the field of mathematics education by expanding the application of knowledge transfer, moving it from only a forward focused direction to include, also, a backward focused direction. An advisory board of scholars with expertise in mathematics education, assessment, social interactions, quantitative reasoning and measurement will support the project. The research will occur in diverse classrooms and result in presentations at the annual conferences of national organizations, peer-reviewed publications, as well as a website for teachers which will explain both the theoretical model and the findings from the project. An undergraduate university course and professional development workshops using video data from the project are also being developed for pre-service and in-service teachers. Ultimately, the research findings will generate new knowledge and offer guidance to elementary school teachers as they prepare their students for algebra.

The research involves three phases. The first phase includes observations and recordings of four Algebra I classrooms and will test students' understanding of linear functions before and after the lessons on quadratic functions. This phase will also include interviews with students to better understand their reasoning about linear function problems. The class sessions will be coded for the kind of reasoning that they promote. The second phase of the project will involve four cycles of design research to create quadratic and linear function activities that can be used as instructional interventions. In conjunction with this phase, pre-service teachers will observe teaching sessions through a course that will be offered concurrently with the design research. The final phase of the project will involve pilot-applied research which will test the effects of the instructional activities on students' linear function reasoning in classroom settings. This phase will include treatment and control groups and further test the hypotheses and instructional products developed in the first two phases.

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.

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.

Learning about Ecosystems Science and Complex Causality through Experimentation in a Virtual World

This project will develop a modified virtual world and accompanying curriculum for middle school students to help them learn to more deeply understand ecosystems patterns and the strengths and limitations of experimentation in ecosystems science. The project will build upon a computer world called EcoMUVE, a Multi-User Virtual Environment or MUVE, and will develop ways for students to conduct experiments within the virtual world and to see the results of those experiments.

Project Email: 
Lead Organization(s): 
Award Number: 
1416781
Funding Period: 
Mon, 09/01/2014 to Thu, 08/31/2017
Full Description: 

EcoXPT from videohall.com on Vimeo.

Comprehending how ecosystems function is important knowledge for citizens in making decisions and for students who aspire to become scientists. This understanding requires deep thinking about complex causality, unintended side-effects, and the strengths and limitations of experimental science. These are difficult concepts to learn due to the many interacting components and non-linear interrelationships involved. Ecosystems dynamics is particularly difficult to teach in classrooms because ecosystems involve complexities such as phenomena distributed widely across space that change over long time frames. Learning when and how experimental science can provide useful information in understanding ecosystems dynamics requires moving beyond the limited affordances of classrooms. The project will: 1) advance understanding of experimentation in ecosystems as it can be applied to education; 2) show how student learning is affected by having opportunities to experiment in the virtual world that simulate what scientists do in the real world and with models; and 3) produce results comparing this form of teaching to earlier instructional approaches. This project will result in a learning environment that will support learning about the complexities of the earth's ecosystem.

The project will build upon a computer world called EcoMUVE, a Multi-User Virtual Environment or MUVE, developed as part of an earlier NSF-funded project. A MUVE is a simulated world in which students can virtually walk around, make observations, talk to others, and collect data. EcoMUVE simulates a pond and a forest ecosystem. It offers an immersive context that makes it possible to teach about ecosystems in the classroom, allowing exploration of the complexities of large scale problems, extended time frames and and multiple causality. To more fully understand how ecosystems work, students need the opportunity to experiment and to observe what happens. This project will advance this earlier work by developing ways for students to conduct experiments within the virtual world and to see the results of those experiments. The project will work with ecosystem scientists to study the types of experiments that they conduct, informing knowledge in education about how ecosystem scientists think, and will build opportunities for students that mirror what scientists do. The project will develop a modified virtual world and accompanying curriculum for middle school students to help them learn to more deeply understand ecosystems patterns and the strengths and limitations of experimentation in ecosystems science. The resulting program will be tested against existing practice, the EcoMUVE program alone, and other programs that teach aspects of ecosystems dynamics to help teachers know how to best use these curricula in the classroom.

Supporting Secondary Students in Building External Models (Collaborative Research: Damelin)

This project will (1) develop and test a modeling tool and accompanying instructional materials, (2) explore how to support students in building and using models to explain and predict phenomena across a range of disciplines, and (3) document the sophistication of understanding of disciplinary core ideas that students develop when building and using models in grades 6-12. 

Lead Organization(s): 
Award Number: 
1417809
Funding Period: 
Fri, 08/01/2014 to Tue, 07/31/2018
Full Description: 

The Concord Consortium and Michigan State University will collaborate to: (1) develop and test a modeling tool and accompanying instructional materials, (2) explore how to support students in building and using models to explain and predict phenomena across a range of disciplines, and (3) document the sophistication of understanding of disciplinary core ideas that students develop when building and using models in grades 6-12. By iteratively designing, developing and testing a modeling tool and instructional materials that facilitate the building of dynamic models, the project will result in exemplary middle and high school materials that use a model-based approach as well as an understanding of the potential of this approach in supporting student development of explanatory frameworks and modeling capabilities. A key goal of the project is to increase students' learning of science through modeling and to study student engagement with modeling as a scientific practice. 

The project provides the nation with middle and high school resources that support students in developing and using models to explain and predict phenomena, a central scientific and engineering practice. Because the research and development work will be carried out in schools in which students typically do not succeed in science, the products will also help produce a population of citizens capable of continuing further STEM learning and who can participate knowledgeably in public decision making. The goals of the project are to (1) develop and test a modeling tool and accompanying instructional materials, (2) explore how to support students in building, using, and revising models to explain and predict phenomena across a range of disciplines, and (3) document the sophistication of understanding of disciplinary core ideas that students develop when building and using models in grades 6-12. Using a design-based research methodology, the research and development efforts will involve multiple cycles of designing, developing, testing, and refining the systems modeling tool and the instructional materials to help students meet important learning goals related to constructing dynamic models that align with the Next Generation Science Standards. The learning research will study the effect of working with external models on student construction of robust explanatory conceptual understanding. Additionally, it will develop a set of professional development resources and teacher scaffolds to help the expanding community of teachers not directly involved in the project take advantage of the materials and strategies for maximizing the impact of the curricular materials.

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.

Developing and Testing the Internship-inator, a Virtual Internship in STEM Authorware System

The Internship-inator is an authorware system for developing and testing virtual internships in multiple STEM disciplines. In a virtual internship, students are presented with a complex, real-world STEM problem for which there is no optimal solution. Students work in project teams to read and analyze research reports, design and perform experiments using virtual tools, respond to the requirements of stakeholders and clients, write reports and present and justify their proposed solutions. 

Award Number: 
1418288
Funding Period: 
Mon, 09/01/2014 to Fri, 08/31/2018
Full Description: 

Ensuring that students have the opportunities to experience STEM as it is conducted by scientists, mathematicians and engineers is a complex task within the current school context. This project will expand access for middle and high school students to virtual internships, by enabling STEM content developers to design and customize virtual internships. The Internship-inator is an authorware system for developing and testing virtual internships in multiple STEM disciplines. In a virtual internship, students are presented with a complex, real-world STEM problem for which there is no optimal solution. Students work in project teams to read and analyze research reports, design and perform experiments using virtual tools, respond to the requirements of stakeholders and clients, write reports and present and justify their proposed solutions. The researchers in this project will work with a core development network to develop and refine the authorware, constructing up to a hundred new virtual internships and a user group of more than 70 STEM content developers. The researchers will iteratively analyze the performance of the authorware, focusing on optimizing the utility and the feasibility of the system to support virtual internship development. They will also examine the ways in which the virtual internships are implemented in the classroom to determine the quality of the STEM internship design and influence on student learning.

The Intership-inator builds on over ten years of NSF support for the development of Syntern, a platform for deploying virtual internships that has been used in middle schools, high schools, informal science programs, and undergraduate education. In the current project, the researchers will recruit two waves of STEM content developers to expand their current core development network. A design research perspective will be used to examine the ways in which the developers interact with the components of the authorware and to document the influence of the virtual internships on student learning. The researchers will use a quantitative ethnographic approach to integrate qualitative data from surveys and interviews with the developers with their quantitative interactions with the authorware and with student use and products from pilot and field tests of the virtual internships. Data-mining and learning analytics will be used in combination with hierarchical linear modeling, regression techniques and propensity score matching to structure the quasi-experimental research design. The authorware and the multiple virtual internships will provide researchers, developers, and teachers a rich learning environment in which to explore and support students' learning of important college and career readiness content and disciplinary practices. The findings of the use of the authorware will inform STEM education about the important design characteristics for authorware that supports the work of STEM content and curriculum developers.

Primary School Organizations as Open Systems: Strategic External Relationship Development to Promote Student Engagement in STEM Topics

This study explores the following issues in 9 schools across 3 neighborhoods: (1) How student engagement in STEM is enabled and constrained by the school's relations with its external community; (2) The similarities and differences in partnerships across different types of schools in three different urban neighborhoods by mapping networks, and assessing the costs and benefits of creating, maintaining, and dissolving network ties; and (3) How to model school and network decisions, relations, and resources using an operations research framework.

Lead Organization(s): 
Partner Organization(s): 
Award Number: 
1344266
Funding Period: 
Tue, 10/01/2013 to Fri, 09/30/2016
Full Description: 

This INSPIRE award is partially funded by the Science of Organization Program in the Division of Social and Economic Sciences in the Social, Behavioral and Economic Sciences Directorate, and the Math and Science Partnership Program and the Discovery Research K-12 program in the Division of Research on Learning in the Education and Human Resources Directorate.

Our country faces a decline in student engagement, particularly in Science, Technology, Engineering, and Mathematics (STEM) disciplines and among underrepresented minority groups. Most often this problem is discussed in the context of an achievement gap, where racial and socioeconomic groups perform unequally on academic assessments. To understand what creates the achievement gap, researchers must understand the STEM "opportunity gap" that exists between students from different backgrounds, where these same students achieve differently because of varying exposure to out-of-school enrichment and learning experiences. The STEM opportunity gap arises from the inequity of out-of-school learning experiences for children. Therefore, efforts to engage minorities and women in STEM in primary schools will only succeed if we consider the complex organizational environment in which primary schools operate. The focus of this study is on what interorganizational relationships are necessary for schools to maintain to ensure equitable, efficient, and effective opportunities for students to engage in STEM. External relationships require schools to commit time and resources, and schools must decide which relationships to develop and maintain. Understanding what kinds of relationships particular school types invest in and what level of effort to commit to maintaining those relationships is important for improving student engagement opportunities in STEM.

Specifically, the study explores the following issues in 9 schools across 3 neighborhoods in Chicago, IL:

(1) How student engagement in STEM is enabled and constrained by the school's relations with its external community.

(2) The similarities and differences in partnerships, particularly STEM-related partnerships, across different types of schools in three different urban neighborhoods by mapping networks, and assessing the costs and benefits of creating, maintaining, and dissolving network ties.

(3) How to model school and network decisions, relations, and resources using an operations research framework. The model prescribes network configurations that address strategic, tactical, and operational concerns, to ensure the school will equitably, efficiently, and effectively utilize partners to improve student engagement in STEM.

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