Learning Progression

Development of the Electronic Test of Early Numeracy

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

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

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

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

Longitudinal Learning of Viable Argument in Mathematics for Adolescents

This project builds on a prior study that demonstrated increases in students' knowledge of argumentation and their performance on mathematics assessments. The project will extend the use of the argumentation intervention into all eighth grade content areas, with a specific focus on students' learning of reasoning and proof, and contribute to understanding how students' learning about mathematical practices that can help them learn mathematics better.

Lead Organization(s): 
Partner Organization(s): 
Award Number: 
1621438
Funding Period: 
Thu, 09/01/2016 to Mon, 08/31/2020
Full Description: 

The project will examine learning in eighth grade mathematics with a specific focus on students' learning of reasoning and proof. The intervention builds on a prior study in algebra that demonstrated increases in students' knowledge of argumentation and their performance on mathematics assessments. This project will extend the use of the argumentation intervention into all eighth grade content areas. The investigators will also address support for teachers in the form of teacher materials that link the argumentation content with mathematics standards and state-wide assessments, and a learning progression to engage students in proving tasks. The project will use assessments of mathematics learning and additional data from teachers and students to understand the impact of the argumentation intervention on teachers and students. The project contributes to understanding how students can learn about mathematical practices, such as proving, that can help them learn mathematics better. A significant contribution will be the definition of aspects of proving and descriptions of student outcomes that can be used to measure how well students have achieved these components of proving. The Discovery Research PreK-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. This project is also supported by NSF's EHR Core Research (ECR) program. The ECR program emphasizes fundamental STEM education research that generates foundational knowledge in the field.

The project suggests twelve conceptual pillars that are combined with classroom processes and assessable outcomes to examine the use of argumentation practices in the teaching of eighth grade mathematics content. The investigation of classroom support for argumentation includes research questions that focus on improvement on state-level assessments, students' ability to construct mathematical arguments, and the conceptual progression that supports students' understanding of argumentation and proof. In addition, the study will examine teachers' role in argumentation in the classroom and their perception of potential challenges for classroom implementation. The study will use an experimental design to examine an intervention for mathematical reasoning and proof in eighth grade. The project includes a treatment group of teachers that will participate in professional development including a summer institute followed by instructional coaching over a two year period.

Development and Empirical Recovery for a Learning Progression-Based Assessment of the Function Concept

The project will design an assessment based on learning progressions for the concept of function - a critical concept for algebra learning and understanding. The goal of the assessment and learning progression design is to specifically incorporate findings about the learning of students traditionally under-served and under-performing in algebra courses.

Lead Organization(s): 
Award Number: 
1621117
Funding Period: 
Thu, 09/15/2016 to Mon, 08/31/2020
Full Description: 

The project will design an assessment based on learning progressions for the concept of function. A learning progression describes how students develop understanding of a topic over time. Function is a critical concept for algebra learning and understanding. The goal of the assessment and learning progression design in this project is to specifically incorporate findings about the learning of students traditionally under-served and under-performing in algebra courses. The project will include accounting for the social and cultural experiences of the middle and high school students when creating assessment tasks. The resources developed should impact mathematics instruction (especially for algebra courses) by creating a learning progression which captures the range of student performance and appropriately places them at distinct levels of performance. The important contribution of the work is the development of a learning progression and related assessment tasks that account for the experiences of students often under-served in mathematics. 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 learning progression development will begin by comparing and integrating existing learning progressions and current research on function learning. This project will develop an assessment of student knowledge of function based on learning progressions via empirical recovery (looking for the reconstruction of theoretical levels of the learning theory). Empirical recovery is the process through which data will be collected that reconstruct the various levels, stages, or sequences of said learning progression. The development of tasks and task models will include testing computer-delivered, interactive tasks and rubrics that can be used for human and automated scoring (depending on the task). Item response theory methods will be used to evaluate the assessment tasks' incorporation of the learning progression.

Supports for Elementary Teachers Implementing the NGSS: Challenges and Opportunities across Science, Technology, and Engineering

STEM Categorization: 
Day: 
Fri

Consider methods and challenges associated with supporting upper elementary teachers’ implementation of NGSS-based classroom interventions in this structured poster session.

Date/Time: 
9:15 am to 10:45 am
Session Materials: 

In this structured poster session, a set of projects will present and discuss resources, models, and tools (RMTs) designed to support upper elementary teachers to implement an array of curricular and instructional interventions reflecting diverse disciplinary concepts and practices embodied in NGSS. The session aims to provide a forum for exploring diverse approaches to improving science in 3rd-5th-grade classrooms and engage in discussion about how these ideas can advance systemic efforts to support quality science instruction and student learning. 

Session Types: 

Scientific Modeling across the K–12 Continuum: Alignment between Theoretical Foundations and Classroom Interventions

STEM Categorization: 
Day: 
Thu

Explore methods and challenges associated with supporting and evaluating scientific modeling in K–12 classrooms in this structured poster session.

Date/Time: 
2:15 pm to 3:45 pm
Session Materials: 

In this interactive panel symposium, presenters will draw from a set of active DR K-12 projects to explore a diverse array of resources, models, and tools (RMTs) designed to operationalize varying perspectives on scientific modeling in elementary, middle, and secondary classrooms across disciplinary domains.

Session Types: 

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

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.

Teaching Viable Argumentation and Measuring the Effects

Day: 
Tues

How do we encourage referent-based mathematical argumentation without encouraging students to request that examples accompany otherwise viable arguments? Assessment concerns are explored and discussed.

Date/Time: 
1:45 pm to 3:45 pm
2014 Session Types: 
Feedback Session (Work in Development)
Presenters: 

The LAMP project has developed a sequence of lessons in a hypothetical learning trajectory that targets students’ ability to write viable arguments in algebraic contexts. Most of the lessons encourage students to produce a referent (e.g., variable expression or equation, generic example, diagram) as the foundation of their argument. Students come to the lessons with a predisposition for example production in support of their claims and to augment arguments.

CAREER: Investigating Differentiated Instruction and Relationships Between Rational Number Knowledge and Algebraic Reasoning in Middle School

The proposed project initiates new research and an integrated education plan to address specific problems in middle school mathematics classrooms by investigating (1) how to effectively differentiate instruction for middle school students at different reasoning levels; and (2) how to foster middle school students' algebraic reasoning and rational number knowledge in mutually supportive ways.

Lead Organization(s): 
Partner Organization(s): 
Award Number: 
1252575
Funding Period: 
Thu, 08/01/2013 to Tue, 07/31/2018
Full Description: 

Middle school mathematics classrooms are marked by increasing cognitive diversity and students' persistent difficulties in learning algebra. Currently middle school mathematics instruction in a single classroom is often not differentiated for different thinkers, which can bore some students or overly challenge others. One way schools often deal with different thinkers at the same grade level is by tracking, which has also been shown to have deleterious effects on students, both cognitively and affectively. In addition, students continue to struggle to learn algebra, and increasing numbers of middle school students are receiving algebra instruction. The proposed project initiates new research and an integrated education plan to address these problems by investigating (1) how to effectively differentiate instruction for middle school students at different reasoning levels; and (2) how to foster middle school students' algebraic reasoning and rational number knowledge in mutually supportive ways. Educational goals of the project are to enhance the abilities of prospective and practicing teachers to teach cognitively diverse students, to improve doctoral students' understanding of relationships between students' learning and teachers' practice, and to form a community of mathematics teachers committed to on-going professional learning about how to differentiate instruction.

Three research-based products are being developed: two learning trajectories, materials for differentiating instruction developed collaboratively with teachers, and a written assessment to evaluate students' levels of reasoning. The first trajectory, elaborated for students at each of three levels of reasoning, focuses on developing algebraic expressions and solving basic equations that involve rational numbers; the second learning trajectory, also elaborated for students at each of three levels of reasoning, focuses on co-variational reasoning in linear contexts. In addition, the project investigates how students' classroom experience is influenced by differentiated instruction, which will allow for comparisons with research findings on student experiences in tracked classrooms. Above all, the project enhances middle school mathematics teachers' abilities to serve cognitively diverse students. This aspect of the project has the potential to decrease opportunity gaps. Finally, the project generates an understanding of the kinds of support needed to help prospective and practicing teachers learn to differentiate instruction.

The project advances discovery and understanding while promoting teaching, training, and learning by (a) integrating research into the teaching of middle school mathematics, (b) fostering the learning of all students by tailoring instruction to their cognitive needs, (c) partnering with practicing teachers to learn how to implement this kind of instruction, (d) improving the training of prospective mathematics teachers and graduate students in mathematics education, and (e) generating a community of mathematics teachers who engage in on-going learning to differentiate instruction. The project broadens participation by including students from underrepresented groups, particularly those with learning disabilities. Results from the project will be broadly disseminated via conference presentations; articles in diverse media outlets; and a project website that will make project products available, be a location for information about the project for the press and the public, and be a tool to foster teacher-to-teacher communication.

Spatial Mathematics, Engineering, and Science: Toward an Integrated STEM Education

The goal of this project is to develop a provisional learning progression spanning grades K-5 that articulates and tests the potential of experiencing, describing, and representing space as the core of an integrated STEM education. The science of space has an extensive scope within and across disciplinary boundaries of science, mathematics and engineering; the project will create a coherent approach to elementary instruction in which mathematical reasoning about space is systematically cultivated.

Lead Organization(s): 
Award Number: 
1252875
Funding Period: 
Mon, 10/01/2012 to Mon, 09/30/2013
Full Description: 

The goal of Spatial Mathematics, Engineering, and Science: Toward an Integrated STEM Education is to develop a provisional learning progression spanning grades K-5 that articulates and tests the potential of experiencing, describing, and representing space as the core of an integrated STEM education. The science of space has an extensive scope within and across disciplinary boundaries of science, mathematics and engineering, the project will create a coherent approach to elementary instruction in which mathematical reasoning about space is systematically cultivated. Simultaneously, researchers are exploring the potential of spatial mathematics as a resource for engineering design of kinematic machines and for the development of mechanistic reasoning about the behavior of these machines. Work across these disciplines situates and motivates the mathematical work and also provides opportunities to investigate the intersections and contrasts among signature disciplinary practices, such as definition and proof in mathematics, design in engineering, and modeling in science. The research and development is being conducted in a middle school which is a full partner in the project.

In partnership, researchers and participating teachers are designing and implementing curricular approaches intended to support spatial knowledge and reasoning. Professional development will enhance and capitalize on teachers' roles as specialists in student thinking. The research consists of design studies conducted in 12 participating classrooms, K-5, and small-scale teaching experiments conducted with children across the same grade span. The research will establish provisional pathways and landmarks in learning about space, as well as the curricular activities and teacher practices necessary to support integrated STEM learning.

The project is novel in three ways. First, it provides children with early and systematic access to multiple geometries (e.g., plane, cylinder, sphere) to develop sophisticated understandings of powerful, yet experientally accessible concepts, such as straight, and STEM-related practices, such as model, definition and proof. Second, both the National Research Council Science/Engineering and the Common Core State Standards Mathematics highlight the role of practices in the development of disciplinary knowledge, and this project is providing a practical avenue for coordinating the co-development of disciplinary practices and knowledge. Third, the unifying theme of space is threaded through problems and contexts in mathematics, science and engineering, which provide a sound basis for generative STEM integration-integration that does not lose sight of the distinctive practices in different disciplines, but, instead, leverages these distinctions to produce multiple ways of knowing about space. Research and development is being conducted with underrepresented populations of students who are typically underserved in STEM education. Although the numbers of students reached in this phase of the work are relatively modest, the longer-term potential is great, because instruction anchored in space may be more accessible to students who struggle with traditional forms of mathematics education. The increased attention to integrated STEM education at the national level also ensures that this effort is likely to contribute to the knowledge base required to advance interdisciplinary forms of schooling.

The Role and Use of Examples in Learning to Prove

This research project is an investigation of the role that examples play in helping learners become proficient in proving mathematical conjectures. Researchers are building a framework that characterizes the development of example use as students advance from middle school into post secondary school. Using this developmental information, the researchers are creating instructional strategies that help students think about the nature and value of proof as well as how to construct a mathematical proof.

Award Number: 
1220623
Funding Period: 
Mon, 10/01/2012 to Sat, 09/30/2017
Full Description: 

This research project is an investigation of the role that examples play in helping learners become proficient in proving mathematical conjectures. Researchers at the University of Wisconsin and New York University are building a framework that characterizes the development of example use as students advance from middle school into post secondary school. Using this developmental information, the researchers are creating instructional strategies that help students think about the nature and value of proof as well as how to construct a mathematical proof.

The researchers are interviewing middle school students, high school students, undergraduate mathematics majors in college, and practicing mathematicians in order to learn how they use examples in the process of creating a proof or a deductive argument. They are using teaching experiments to test various strategies on individual students and to learn more about students' thinking about the process of proving. They are also using small group instruction to experiment with instructional strategies and move closer to a future goal of preparing materials for full class instruction.

Creating and understanding mathematical proofs has always been difficult for students, and yet it is a critical foundation for developing mathematical understanding. Students often cling to the idea that a large set of examples is sufficient for proving a conjecture true. This misconception has often discouraged instructors from using examples in teaching students to prove conjectures. However, examples can stimulate thinking that guides a student to construct a proof through valid, deductive reasoning. This project is advancing understanding of how students learn to prove and providing tested, instructional strategies that teachers can use to help students.

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