Learning Progression

CAREER: L-MAP: Pre-service Middle School Teachers' Knowledge of Mathematical Argumentation and Proving

This program of research will examine how middle school pre-service teachers' knowledge of mathematical argumentation and proving develops in teacher preparation programs. The project explores the research question: What conceptions of mathematical reasoning and proving do middle school preservice teachers hold in situations that foster reasoning about change, proportionality, and proportional relationships, as they enter their mathematics course sequence in their teacher preparation program, and how do these conceptions evolve throughout the program?

Lead Organization(s): 
Award Number: 
1350802
Funding Period: 
Tue, 07/15/2014 to Tue, 06/30/2020
Full Description: 

The field of mathematics teacher education needs a strong understanding of pre-service teachers' knowledge about the practice of mathematical argumentation and proof, including the development of this knowledge, to effectively move pre-service teachers toward a more sophisticated understanding and enactment of this practice with their own students. The integrated research and educational activities will contribute to the knowledge base teacher education programs need to effectively prepare middle school teachers for meeting the challenges of how to make reasoning and proof an integral aspect of instructional practice. The research results have the potential to guide teacher educators and educational researchers concerned with strengthening pre-service teachers' ability to make reasoning and proving an integral aspect of school mathematics. Consequently, pre-service teachers will be better equipped to develop mathematical reasoning skills in their future students and to support their students in learning mathematics with understanding. Given this country's growing need for a competent STEM workforce, helping all students learn mathematics in a way that supports deeper understanding is a priority. 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.

The objective of this program of research is to examine how middle school pre-service teachers' knowledge of mathematical argumentation and proving develops in teacher preparation programs. The project explores the research question: What conceptions of mathematical reasoning and proving do middle school preservice teachers hold in situations that foster reasoning about change, proportionality, and proportional relationships, as they enter their mathematics course sequence in their teacher preparation program, and how do these conceptions evolve throughout the program? This development will be studied along three dimensions: (a) pre-service teachers' own ability to formulate mathematical arguments, (b) their ability to analyze mathematical arguments, and (c) their ability to analyze situations that engage students in mathematical argumentation and proving. Cross-sectional and longitudinal studies of 60 pre-service teachers' models, or systems of interpretation, of mathematical argumentation and proof in curricular context that foster reasoning about change, proportionality and proportional relationships will be conducted to provide an understanding of the trajectory that captures how pre-service teachers develop their knowledge of mathematical argumentation and proving throughout their university mathematics preparation program and into their student teaching.

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: Leveraging Contrasting Cases to Investigate Integer Understanding

Most students learn about negative numbers long after they have learned about positive numbers, and they have little time or opportunity to build on their prior understanding by contrasting the two concepts. The purpose of this CAREER project is to identify language factors and instructional sequences that contribute to improving elementary students' understanding of addition and subtraction problems involving negative integers. 

Lead Organization(s): 
Award Number: 
1350281
Funding Period: 
Thu, 05/15/2014 to Sat, 04/30/2022
Full Description: 

Currently, most students learn about negative numbers long after they have learned about positive numbers, and they have little time or opportunity to build on their prior understanding by contrasting the two concepts. Therefore, they struggle to make sense of negative integer concepts, which appear to conflict with their current understanding. The purpose of this CAREER project is to identify language factors and instructional sequences that contribute to improving elementary students' understanding of addition and subtraction problems involving negative integers. A second objective is to identify how elementary teachers interpret their students' integer understanding and use research findings to support their teaching of these concepts. This project is expected to contribute to theories regarding the development of integer understanding as well as what makes a useful contrasting case when learning new, related concepts. Moreover, the results of this project can contribute to our understanding of how to build on students? prior number knowledge rather than contradict it.

The principal investigator will conduct a series of four experimental studies involving a preparation for learning component with students randomly assigned to treatment or control groups. Study 1 will involve second and fourth graders and will test the language factors that support students' understanding of integers. Studies 2-4 will involve second and fifth graders and will test the optimal order in which integer addition and subtraction problems are presented in contrast with each other versus sequentially without contrasts. Using items that measure students? understanding of integers and integer operations, the PI will compare students' gains from pre-tests to post-tests between groups. Further, the investigator will qualitatively code students? solution strategies based on follow-up interviews and written work for additional information on the differences between groups. Following the experimental studies, the PI will work with elementary teachers over three lesson study cycles, during which teachers will implement instruction based on the prior studies? results. The PI will compare the performance of students who participate in the lesson study unit versus control classrooms to measure impact of the unit.

Videos of the lesson study unit, as well as the negative integer lesson plans will be made available for other teachers and teacher educators to use. Further, the investigator will incorporate the research results into an undergraduate mathematics methods course. To ensure that the results of this research reach a wider audience, the investigator will create an integer game and storybook, illustrating key concepts identified through the research, that parents can explore together with their children during family math nights and at home. On a broader scale, this project has the potential to illuminate ways to develop more coherence in the sequencing of mathematics topics to more effectively build on students? current understanding.

Science in the Learning Gardens (SciLG): Factors that Support Racial and Ethnic Minority Students’ Success in Low-Income Middle Schools

Science in the Learning Gardens (SciLG) designs and implements curriculum aligned with Next Generation Science Standards (NGSS) and uses school gardens as learning contexts in grade 6 (2014-2015), grade 7 (2015-2016) and grade 8 (2016-2017) in two low-income urban schools. The project investigates the extent to which SciLG activities predict students’ STEM identity, motivation, learning, and grades in science using a theoretical model of motivational development.

Lead Organization(s): 
Partner Organization(s): 
Award Number: 
1418270
Funding Period: 
Mon, 09/01/2014 to Thu, 08/31/2017
Full Description: 

Science in the Learning Gardens (SciLG) will use school gardens as the context for learning at two low-income middle schools with predominantly racial and ethnic minority students in Portland, Oregon. There are thousands of gardens flourishing across the country that are underutilized as contexts for active engagement in the middle grades. School gardens provide important cultural contexts while addressing environmental and food issues. SciLG will bring underrepresented youth into gardens at a critical time in their intellectual development to broaden the factors that support motivation to pursue STEM careers and educational pathways. The project will adapt, organize, and align two disparate sets of existing resources into the project curriculum: 6th grade science curriculum resources, and garden-based lessons and units. The curriculum will be directly aligned with the Next Generation Science Standards (NGSS). 

The project will use a design-based research approach to refine instruction and formative assessment, and to investigate factors for student success in science proficiency and their motivational engagement in relation to the garden curriculum. The curriculum will be pilot-tested during the first year of the project in five sixth-grade classes with 240 students in Portland Public Schools. Students will be followed longitudinally in grades 7 and 8 in years 2 and 3 respectively, as curricular integration continues. The research team will support participating teachers each year in using their schools' gardens, and study how this context can serve as an effective pedagogical strategy for NGSS-aligned science curriculum. Academic learning will be measured by assessments of student progress towards the end of middle-school goals defined by NGSS. Motivation will be measured by a validated motivational engagement instrument. SciLG results along with the motivational engagement instrument will be disseminated widely through a variety of professional networks to stimulate implementation nationwide.

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.

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.

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.

Teaching STEM with Robotics: Design, Development, and Testing of a Research-based Professional Development Program for Teachers

Using design-based research, with teachers as design partners, the project will create and refine project-based, hands-on robotics curricula such that science and math content inherent in robotics and related engineering design practices are learned. To provide teachers with effective models to capitalize on robotics for elucidating science and math concepts, a design-based Professional Development program will be built using principles of technological, pedagogical, and content knowledge (TPACK).

Lead Organization(s): 
Award Number: 
1417769
Funding Period: 
Mon, 09/01/2014 to Fri, 08/31/2018
Full Description: 

Offering meaningful and motivating engineering contexts, such as robotics, within science and math courses constitutes a compelling strategy to address the Next Generation Science Standards and the Common Core State Standards for Math while enhancing science and math learning for all students. Using design-based research, with teachers as design partners, the project will create and refine project-based, hands-on curricula such that science and math content inherent in robotics and related engineering design practices are learned. To provide teachers with effective models to capitalize on robotics for elucidating science and math concepts, a design-based Professional Development program will be built using principles of technological, pedagogical, and content knowledge (TPACK). To ensure that teachers are well prepared, research-based practices and features of effective Professional Development will be adopted. Experts in robotics, engineering, education, curriculum design, and assessment--with experience in K-12 education, training, and outreach--have formed an interdisciplinary team to make robotics central to and sustainable in middle school science and math classrooms.

The research questions addressed in this project are qualitative in nature as appropriate for design research questions. The methodologies include teacher needs assessment, teachers' perceptions of robotics, pre and post testing, classroom observations, and surveys. Examples of the research questions are:

What characteristics of robotics promote effective learning of middle school science and math?

What elements of Professional Development engender teachers' TPACK of robotics and link it with classroom science and math?

What are student prerequisites to effectively use robotics in science and math learning?

What are the gains in students' STEM engagement, interest, persistence, and career awareness?

The robotics curriculum will include physical science used in robot performance expectations and motion stability. Additionally the curriculum will include the engineering design process consisting of problem definition, solution development, and design improvement. Robotics provides opportunities to support science and engineering practices of the Next Generation Science Standards such as developing and using models, planning and conducting investigations, designing solutions, and analyzing and interpreting data. The project will be aimed at middle school students and will provide substantial teacher professional development to implement the new curriculum modules. The partner schools have student bodies drawn from a diverse student population in New York City.

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.

Improving Formative Assessment Practices: Using Learning Trajectories to Develop Resources That Support Teacher Instructional Practice and Student Learning in CMP2

The overarching goal of this project is to develop innovative instructional resources and professional development to support middle grades teachers in meeting the challenges set by college- and career-ready standards for students' learning of algebra.

Partner Organization(s): 
Award Number: 
1316736
Funding Period: 
Tue, 10/01/2013 to Sat, 09/30/2017
Full Description: 

The overarching goal of this project is to develop innovative instructional resources and professional development to support middle grades teachers in meeting the challenges set by college- and career-ready standards for students' learning of algebra. This 4-year project includes three major components: (1) development and empirical testing of learning trajectories for linear functions and linear equations, (2) collaborations with teachers of Connected Mathematics Project 2 (CMP2) to create and test a set of instructional resources focused on formative assessment processes, and (3) iterative refinement of a professional development model for engaging teachers with the instructional resources in ways that optimize students' learning of algebra. The professional development activities provide opportunities for teachers to develop specialized content knowledge of learning trajectories for linear functions and equations in algebra, processes for interpreting students' performances with respect to those trajectories and providing feedback and additional instructional activities based on "where" the student is with respect to the overall learning trajectory. Such changes in teacher knowledge and practice are anticipated to produce improved student learning outcomes for key concepts and procedures in algebra. One of the major stumbling blocks to teachers' implementation of effective formative assessment practice is the sheer volume and management of the information needed to monitor and interpret student performance. The project addresses this impediment by employing the ASSISTments platform, a web-based online system for delivering mathematics problem sets and capable of adapting problem presentation to student performance in real time.

Research on learning trajectories in mathematics has mostly centered on concepts at the primary school level. While research at this level has been prolific and informative in multiple aspects of mathematics education, there are major knowledge gaps in our understanding of learning trajectories in several domains of mathematics, specifically in algebra. Indeed, there is a growing need for new research and development projects to fill these critical knowledge gaps.

This project focuses on two critical areas in mathematics: students' understanding of linear functions and linear equations, and students' ability to use them to solve problems. Empirically validated learning trajectories will support curriculum development in these areas. In addition, this project contributes to the research base to improve the curriculum standards by providing empirical evidence for hypothesized trajectories for selected content standards for middle school students. Finally, the use of CMP2 augmented by the online management system increases the probability of widespread impact of the professional development model targeted at teachers' formative assessment practices. Although we are using a specific curriculum program, the treatment of linear functions and equations topics in CMP is consistent with other functions-based curricula in the U.S. Thus, the work done in the context of this project will be useful in examining learning trajectories and formative assessment in other instructional programs.

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