Learning Progressions and Trajectories

child climbing a climbing wallThis Spotlight features research on science and math learning progressions and trajectories. The projects share their approaches to addressing issues of equity and describe how they prepare and support teachers in understanding and using learning progressions and trajectories in their practice.

In the Spotlight's blog post, Jere Confrey and Alan Maloney discuss the elements of learning trajectory research that are critical to supporting equity.

In this Spotlight:

Blog Post | Learning Trajectories and Equity: Making a Strong Link Stronger

Jere Confrey, Distinguished University Professor of Mathematics Education, Emerita, North Carolina State University; President, The Math Door,
Alan Maloney, Vice-President, The Math Door

Jere Confrey and Alan MaloneyThe idea of a “hypothetical learning trajectory” proposed by Simon (1995) was a description of a tentative plan to help move students from naive to more sophisticated thinking about a target concept. It was proposed to provide guidance for teachers as they observed students’ responses to tasks that invited and challenged them to engage with increasingly complex ideas. The assumption in that work was that a learning trajectory (LT) was valuable–though always tentative–for imagining and planning instruction in advance. It helped teachers to hypothesize how students might encounter and solve increasingly difficult aspects of a complex idea. Key to that progression was an understanding of what propels conceptual depth—an idea Piaget described as “genetic epistemology” (Piaget, 1970). Genetic epistemology implies that the meaning of an idea unfolds and becomes enriched as one sees various aspects of it, and specifically recognizes how it can be refined to make sense of broader related tasks and deeper connections. It recognizes that the constructive process evolves gradually. Fundamentally, it acknowledges that early reasoning is almost always partial–solving part of the problem space–and may misrepresent or ignore other parts, and thus requires further accommodations, revisions, or distinctions, to become complete. The field has now progressed to develop and synthesize research-based learning trajectories that describe patterns in students’ responses to solving challenging tasks as they advance from naive to sophisticated reasoning about target concepts...Read more.

Featured Projects


A Window into Students' Experiences (AWISE): Developing Mathematics Teachers' Responsive Pedagogies for Linguistically Marginalized Students

PI: Samantha Marshall Pham
STEM Discipline: Mathematics
Grade Levels: 6-12

Project Description: Using design-based research, we will design, implement, investigate, and iteratively refine a video-based coaching model to develop mathematics teachers’ responsive pedagogies for linguistically marginalized students. Building from cutting edge research on linguistically responsive mathematics pedagogies, this project addresses the persistent need to foster mathematics teachers’ learning about supporting linguistically marginalized students. Our novel model centers the experiences of students through video clips as rich tools for teacher learning, and builds learning trajectories for teachers through 3 years of professional development based on teachers’ unique challenges and existing conceptions of responsive pedagogies. Our approach builds from key findings from prior studies: that video-based coaching can support teachers in learning justice-oriented pedagogies such as social justice mathematics and learning to disrupt racialized patterns of exclusion in mathematics classrooms by supporting teachers’ sensemaking about their own students’ unique experiences in mathematics classrooms and giving timely, formative feedback as teachers encounter problems of practice (Horn & Garner, 2022). Central to our model is this core insight: that classroom video holds potential for supporting teacher learning of responsive pedagogies because of its opening of a window into students’ experiences. Our overarching research question is: How do secondary mathematics teachers learn about supporting linguistically marginalized students? The primary outcomes of this research include: a portrait of the challenges and opportunities that mathematics teachers face in supporting linguistically marginalized students, an iteratively refined model of professional development for teachers’ learning of responsive pedagogies, and an empirically-grounded theory of teachers’ learning to support linguistically marginalized students.

How does your project consider issues of equity when designing and/or assessing a learning trajectory?: Equity is at the core of our design: our learning trajectories for teachers are entirely focused around making mathematical learning experiences more equitable for linguistically marginalized students. Dominant approaches to teaching linguistically marginalized students in the U.S. tend to operate under deficit logics (Marshall & Rivera, 2023), despite that research repeatedly documents the advantages of responsive, asset-based pedagogies. We design learning experiences for teachers that support them in responding to students’ assets, responding to student thinking, and responding to students’ cultures, in order to equip them with the skills to teach linguistically marginalized students in more ambitious and equitable ways. We draw on intersectionality theory, critical race theory, and raciolinguistics in devising these learning trajectories for teachers (Marshall & Rivera, 2023), keeping equity for marginalized students in the forefront of all our design efforts.

How can we better prepare and support teachers to understand and effectively utilize learning progressions in their classrooms?: We seek to better support teachers to implement effective learning strategies in their classrooms by organizing their learning around rich representations of practice such as classroom video (Horn & Garner, 2022). We conjecture that in order to respond to who students are and the assets that they bring to mathematical learning, teachers need to gain understanding of students’ experiences in the classroom. By capturing student-to-student discourse, we can provide a window into students’ experiences and interactions that is typically inaccessible to teachers as they reflect on their pedagogy (Borko et al., 2011). In Marshall’s previous study, she found that such representations of practice could serve as ripe resources for mathematics teachers’ learning of justice-oriented pedagogies (Marshall, 2020, 2022). These findings inform our conjecture that teacher learning activities should support teachers’ pedagogical sensemaking around their own students and their experiences in the mathematics classroom. Indeed, a central challenge of professional development on responsive pedagogies is that such pedagogies must respond to particular students — who they are and what assets they bring (Yeh, 2021). Therefore, to support teachers’ development of responsive pedagogies, it stands to reason that teacher learning experiences should center students.

Initial findings: We are in the initial stages of data collection, but we will have findings to share related to mathematics teachers’ current conceptions about teaching linguistically marginalized students soon.


Website: https://research.ced.ncsu.edu/awise/


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Developing and Testing a Learning Progression for Middle School Physical Science Incorporating Disciplinary Core Ideas, Science and Engineering Practices, and Crosscutting Concepts

PI: Peng He
STEM Discipline: Physical Sciences
Grade Levels: 6-8

Project Description: This three-year project develops and tests three-dimensional learning progressions (3DLP) for middle school physical science that incorporate the three dimensions of scientific knowledge from the Framework for K-12 Science Education: Disciplinary Core Ideas of Matter and its Interaction, and Energy; the Science and Engineering Practices of Constructing Explanations and Developing and Using Models; and the Crosscutting Concepts of Cause and Effect and Systems and System Models. To achieve the project goals, we employ a design-based research approach to iteratively design and test our 3DLPs for middle school physical sciences. The revision and testing process uses various data sources, including expert feedback, teacher and student interviews, classroom observations, and teacher and student artifacts. Our 3DLPs provide multiple learning pathways to support students from diverse backgrounds, especially those with low socioeconomic status and underrepresented populations, to develop a more sophisticated and deeper understanding of middle school physical sciences. Participating teachers will receive professional learning and guidelines on using the 3DLPs to adapt their local curriculum and instruction materials. The project will investigate how teachers use the 3DLPs to improve their teaching to support student learning and monitor students’ growth in the 3DLPs aligned classrooms.

Our approach advances the development of learning progressions in two ways: first, we use an evidence-centered design approach to develop fine-grained size 3DLPs that guide teachers' adaptation of their local materials to align with the NGSS Performance Expectations. We partner with teachers to provide professional learning, focusing on how learning develops across time. Second, we advance the existing learning progression frameworks by articulating multiple learning pathways between levels of 3DLP to support learners with diverse backgrounds.

How does your project consider issues of equity when designing and/or assessing a learning trajectory?: Our project addresses equity issues by developing and testing 3DLPs with multiple learning pathways between levels to support students from diverse backgrounds, particularly minoritized and marginalized racial and ethnic groups, to support their learning. We co-developed our 3DLPs with urban, rural, and fringe school district teachers. In our 3DLPs, we articulated multiple learning pathways to provide diverse learning opportunities for students to advance from low to upper levels in the learning progressions, supporting learning to develop more sophisticated understandings. The design of our 3DLPs guides teachers' adaptations of their local materials so that students can experience different learning opportunities along various pathways. In addition, we monitor focal groups of students' learning pathways across time to uncover the mechanisms of using our 3DLPs to advance equitable student learning in science classrooms.

How can we better prepare and support teachers to understand and effectively utilize learning progressions in their classrooms?: We provided mentorships for our participating teachers in two ways: co-designing and principle-based extended professional learning. First, in our co-design phase, we explicitly explained the importance of our 3DLP framework based on the developmental perspectives of learning and the notion of usable knowledge. Participating teachers and our research team work together to develop and revise the 3DLPs and adapt their local materials to 3DLPs. Second, we provide in-depth professional learning over two years as a model for how knowledge develops over time. We hope that our modeling will allow teachers to understand better the development perspective of learning to support the implementation of our 3DLPs in their science classrooms. Through co-designing and participating in our extended professional learning, teachers will better understand the developmental perspective of learning, the 3DLPs, and how to use 3DLPs in their classrooms.

What are the anticipated next steps in advancing research on this specific learning progression?: Our anticipated next steps in our 3DLP research will focus on three areas: 1) Automatically analyze individual learner responses to 3-dimensional assessment tasks to track their development over time; 2) Provide timely, personalized teacher and student feedback; and 3) Develop 3DLP across grade band. First, we look to tackle core methodology issues that have made it challenging to develop and verify the effectiveness of 3DLPs to support students' better learning across time. As learning is a developmental process, embedded classroom-based performance assessments along with 3DLPs should drive classroom experiences and assessment over time by providing feedback and instructional support. However, monitoring and analyzing students’ progress along 3DLPs across grade bands is challenging. Identifying students’ challenges from their responses to drawn models and written explanations would help teachers adapt their instruction to meet the needs of students in their classrooms. Yet, analyzing models in a timely fashion provides challenges for teachers. We anticipate that innovative technologies (e.g., Generative Artificial Intelligence (AI)) can assist us in automatically analyzing and identifying students’ challenges on scientific explanation and modeling tasks. Second, we will explore how generative AI can deliver personalized feedback to teachers and students in a timely manner. Providing teachers with research-based instructional strategies, along with 3DLPs, will help move students to the next levels, supporting students in developing more sophisticated understanding. Third, we need to develop 3DLPs in physical science to include the elementary and high grades. Here, too, generative AI tools could support this effort, in automatically analyzing students' responses to test and validate 3DLP. Still, the challenging work will lie in humans' creative and critical thinking to develop the 3DLP and associated assessment tasks.

We view AI as a partner to move forward in student learning. Developing generative AI systems to analyze and provide students and teachers with timely feedback could support student understanding, which is essential to developing student understanding along the 3DLP.  In this way, we hope to reduce the perpetual achievement gap between individuals in various groups.

Initial findings: We developed two 3DLPs of Matter and Its Interaction and Thermal Energy based on feedback from experts and teachers and students’ cognitive interviews. See the 3DLP of Matter and Its Interaction in Figure 1. Initial findings involve exploring students’ understanding of 3D knowledge aligned with our initial 3DLP. We used mixed methods (Creswell & Plano Clark, 2018) to conduct quantitative and qualitative analysis to articulate students’ 3D understanding aligned with our 3DLP levels.

Regarding Thermal Energy, we found that middle school students performed well using the organization and interaction of molecules to explain what happened during phase change but had challenges in understanding average kinetic energy and potential energy at the molecular level. They could implicitly and explicitly use system models to describe a given phenomenon but need support to understand systems and interactions to explain the mechanistic process of a given phenomenon. Regarding Matter and Its Interaction, our results show that middle school students could understand the properties of matter to identify substances and chemical reactions but need more support in understanding atomic arrangements and conservation of mass and atoms. Students could construct descriptive and relational explanations in explaining phenomena related to the topic.

The initial findings above are based on a few data we collected in Year 1. We are collecting and analyzing at least 20 more student interviews to boost our findings of exploring students’ understanding of 3D knowledge aligned with our initial 3DLP.


Website: https://3dlp.org/


  • He, P., Shin, N., & Krajcik, J. (in press). Developing three-dimensional learning progressions of energy, interaction, and matter at middle school level: A design-based research. In Jin, H., Yan, D., & Krajcik, J. Handbook of Research in Science Learning Progressions. Routledge
  • He, P. Shin, N. Kaldaras L., & Krajcik, J. (in press). Integrating artificial intelligence into learning progression-based learning systems to support student knowledge-in-use: Opportunities and challenges. In Jin, H., Yan, D., & Krajcik, J. Handbook of Research in Science Learning Progressions. Routledge
  • He, P., Zhai, X., Shin, N., Krajcik, J. (2023). Applying Rasch measurement to assess knowledge-in-use in science education. In: Liu, X., Boone, W.J. (eds) Advances in Applications of Rasch Measurement in Science Education. Contemporary Trends and Issues in Science Education, vol 57. Springer, Cham. https://doi.org/10.1007/978-3-031-28776-3_13.


  • He, P., Shin, N., & Krajcik, J. (accepted). Developing three-dimensional learning progressions of energy, interaction, and matter at middle school level: A design-based research. Paper proposal submitted to the 97th NARST Annual International Conference, Denver, CO.
  • He, P., Shin, N., & Krajcik, J. (accepted). Using Generative AI to Automatically Identify Students' Three-Dimensional Understanding in an NGSS-Aligned Learning Progression. Paper proposal submitted to the 97th NARST Annual International Conference, Denver, CO.
  • He, P. Shin, N. Kaldaras L., & Krajcik, J. (accepted). Integrating artificial intelligence into learning progression to support student knowledge-in-use: Opportunities and challenges. Paper proposal submitted to the Annual Meeting of the American Educational Research Association, Philadelphia, PA.
  • He, P., Shin, N., Huang, M., Zeng, M., Bowers, J., & Krajcik, J. (June, 2023). Developing and testing a learning progression for middle school physical science incorporating disciplinary core ideas, science and engineering practices, and crosscutting concepts. Poster Presentation at 2023 CADRE PI Meeting. Washington, DC.
  • Huang, M., He, P., Zeng, M., Shin, N., Bowers, J., & Krajcik, J. (accepted). Developing a Three-Dimensional Learning Progression for the Properties and Structure of Matter at the Middle School Level. Paper proposal submitted to the 97th NARST Annual International Conference, Denver, CO.
  • Zeng, M., He, P., Huang, M., Shin, N., Bowers, J., & Krajcik, J. (accepted). Developing a Three-Dimensional Learning Progression for Thermal Energy at Middle School Science. Paper proposal submitted to the 97th NARST Annual International Conference, Denver, CO.

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Evaluating Effects of Automatic Feedback Aligned to a Learning Progression to Promote Knowledge-In-Use

PI: Kevin Haudek
STEM Disciplines: Physical Sciences, Chemistry
Grade Levels: 9-12

Project Description: This project examines the effect of an artificial intelligence (AI) assessment system that automatically generates feedback based on students’ open-ended assessment responses in physical science consistent with previously-developed learning progressions about electrical interactions. The AI-based scoring system will provide individualized feedback to students and class summaries to their teachers as part of formative assessment in an existing high school physical science curriculum aligned with performance expectations in the Next Generation Science Standards to advance student learning.  

The AI-based system applies natural language processing, image recognition, and supervised machine learning to score students’ written explanations and electronically drawn models.  Individualized feedback will be aligned to the learning progression and delivered based on the AI classification of the responses. The project will then examine whether the automatic feedback supports students’ learning outcomes and their development with respect to the learning progression on electrical interactions. The project promotes students’ knowledge of science by engaging them in scientific practices, like modeling, with key disciplinary ideas and using crosscutting concepts to make sense of compelling phenomena and by providing real-time feedback to students. The project addresses two research questions: (1) What is the effect of automatic feedback on student performance along a previously validated learning progression for physical science aligned with the Next Generation Science Standards? (2) What is the effect of automatic feedback on how students connect ideas to advance in learning progression levels?

How does your project consider issues of equity when designing and/or assessing a learning trajectory?: Increased attention has been given to creating inclusive science classrooms that promote equity by broadening access and opportunity for all students. Educators realize that to prepare citizens, science education needs to provide opportunities for all students to learn disciplinary core ideas, crosscutting concepts   and scientific practices to  demonstrate knowledge-in-use. Emerging work in AI and science education has shown potential for promoting equitable science teaching and learning. AI -- such as machine learning, computer vision, and natural language processing -- can support evaluation of complex assessment tasks and provide students with timely information based on their individual or group performance with the aim to promote learning.

This project contributes to designing classroom learning environments that are more equitable and inclusive and by providing timely and substantive feedback to students engaged in authentic scientific practices, like modeling, while solving complex problems or making sense of phenomena.This is especially needed in urban and high-need classrooms where limited teacher time and resources make it challenging to provide timely and appropriate feedback on such tasks. As part of the assessment & feedback development in this project, we collect and evaluate a wide range of student responses from numerous classes and several school districts. This ensures we examine a broad range of real student performances over all levels of a learning progression to train the AI system and to align system outputs to the learning progression.  Since there are multiple ways students can express proficiency within a learning progression level, we hope to capture and identify these different ways of knowing, thus contributing to more equitable assessment practice. Further, we plan to test the automated feedback statements by interviewing teachers from multiple districts with different experiences and students about these potential statements. Specifically, we ask students to interpret feedback statements to a real assessment response and what revisions they likely would make in response to the feedback. This will ensure the feedback is interpretable by students at the grade level and cover a range of common performances on the assessment.

What are the anticipated next steps in advancing research on this specific learning progression?: Our current work focuses on the first two units of a year long physical science program, Interactions. We hope to extend our work to the last two units of Interactions. This will allow us to track student development across a full school year to explore the depth of student learning that occurs when students receive developmental feedback to support their understanding of 3-dimensional knowledge. Learning progressions tied to tasks that make students' thinking visible and which help develop knowledge offer promise to advance student learning. When linked to an AI system that can provide automatic and specific feedback to learners and teachers,learning progression research has the potential to support a wide range of students in our nation’s public schools, with the potential of reducing the achievement gap among students from various backgrounds. 


Website: https://interactionsautofeedback.open3d.science/home


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Exploratory Evidence on the Factors That Relate to Elementary School Science Learning Gains Among English Language Learners

PI: Chris Curran
STEM Discipline: Science and Engineering
Grade Levels: K-5

Project Description: Students who are in the process of adding English to their linguistic repertoires represent an increasing proportion of students in US public schools. This research examines the elementary school science test score trajectories of multilingual students with the nationally representative Early Childhood Longitudinal Study (ECLS). We address the following research questions: How do science test performance trajectories vary across and within MLs student groups in elementary school? Specifically, how do they vary for a) ML students who receive formal EL services at school and those that do not, b) ML students who predominantly speak English in the home and those that predominantly speak another language, c) ML students who are Spanish speakers and ML students who speak less common languages? Our study provides some of the first national evidence on the elementary school science test score trajectories of multilingual learners. In particular, our analyses show that science learning trajectories vary as much between ML subgroups as they do between MLs and non-MLs. Furthermore, we find evidence of significant convergences in ML student performance across the elementary school years, particularly among non-Spanish speaking ML students.

How does your project consider issues of equity when designing and/or assessing a learning trajectory?: Our research demonstrates that efforts to address disparities in science learning must consider both the science learning opportunities and the linguistic support opportunities that multilingual learners experience. We show that even though these learning opportunities are not always equally available to all subgroups, many ML students make significant science learning gains in elementary school, with some ML subgroups surpassing native English speakers. We point to the linguistic and scientific assets that these students bring to the classroom and the lessons we can learn from their success to support science learning more broadly.

How can we better prepare and support teachers to understand and effectively utilize learning progressions in their classrooms?: Our work shows that students, including multilingual learner students, progress in their science learning in very different ways throughout elementary school. It is important for teachers to appreciate and respond to the unique needs of individual students and be aware of the varying trajectories that subgroups of students progress on. Differentiating science instructional practices in ways that meet the linguistic needs of MLs holds promise for improving their science learning.

What are the anticipated next steps in advancing research on this specific learning progression?: Students’ learning trajectories are a function of a wide range of inputs from both schools and other contexts.  Our study examines the science instructional inputs and the language support inputs that schools provide, focusing on the interplay between the two for multilingual learner’s science achievement trajectories in elementary school. These inputs interact with each other and can vary across subgroups of ML students. While our work provides a foundation for identifying promising school practices to support ML student learning, future research can further evaluate interventions based on these promising practices.

Initial findings:

  • Elementary science test score trajectories vary as much across ML subgroups as they do between MLs and non-MLs.
  • There are differences in test score gaps in science compared to math and reading, though not for all subgroups.
  • MLs who speak a language other than Spanish close the science test score gap by the end of elementary school.
  • Science inputs measured in the ECLS are relatively evenly distributed across MLs, non-MLs, and ML subgroups.  Linguistic supports are not.


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Investigating the Role of Collaboration on the Development of Student Ideas Using a Learning Progression for the Function Concept

PI: Edith Aurora Graf
STEM Discipline: Mathematics
Grade Levels: 9-12

Project Description: A feature of The Algebra Project’s Five-Step Curricular Process (Moses & Cobb, 2001; Moses et al., 1989; Moses et al., 2009) is that students discuss mathematics in small teams using their own language to support the transition from a concrete experience to an abstract representation. This approach is consistent with the intent of a learning progression/trajectory (LP/T), which describes the development of student ideas and can be used to guide instruction (see Confrey, 2019, for a review). In our project, we first adapted learning progression-based tasks from our earlier work (supported by NSF grant 1621117) for use in a collaborative problem-solving (CPS) context. The study in which students are engaging consists of four phases: 1) Students respond individually to an extended task assessing the concept of function; 2) Students revisit and respond to the task in small teams, while using an online chat; 3) The teacher leads a classroom discussion about the task; and 4) students respond individually to a parallel task. Some of the student chats are actively facilitated by near-peer and graduate student mentors trained in the use of Talk Moves (Michaels & O’Connor, 2012). Student responses are being scored using the levels of a learning progression. The chat data are being coded through three lenses: a) the lens of the learning progression (Eames et al., 2021; Graf et al., 2021), b) the lens of a Collaborative Problem Solving (CPS) ontology (Andrews-Todd & Forsyth, 2020), and c) the lens of facilitation.

How does your project consider issues of equity when designing and/or assessing a learning trajectory?: Our project considers issues of equity in four ways. First, the organizations contributing to this work are dedicated to ensuring the mathematical literacy of those students least served by traditional methods, with the goal of enabling universal access to advanced mathematical study and STEM careers. We are especially focused on supporting students who may not have had opportunity to demonstrate what they know on traditional achievement tests. Their perspectives are essential to consider in the design and validation of an LP/T. Second, in an online chat in small teams, students may display nuances of mathematical understanding that feed back into the design of the LP/T. Participating in an online chat may also cultivate student agency. Third, the mentors who are facilitating the online chat practice to engage all students in a team. Fourth and finally, we work with teachers to develop a discursive practice and facilitation of student discourse which promotes equity of participation in the mathematics classroom. 

How can we better prepare and support teachers to understand and effectively utilize learning progressions in their classrooms?: In our experience, teachers need the time, space and support to unpack the ideas in the LP/T. As part of our project, we have developed activities linked to the LP/T and the Five-Step Curricular Process. We have engaged in workshops with teachers where they work through an LP/T-based activity in small teams and then discuss with the larger group. We have also shared hypothetical student responses to different tasks.

What are the anticipated next steps in advancing research on this specific learning progression?: Once we have coded the chat through the lens of the LP/T, we expect this will add nuance to the LP/T we have already developed. The revised LP/T can in turn be used as a basis for the design of future formative assessments, including those that involve collaboration. It will also have the potential to be used by teachers for interpreting and guiding classroom discussion.

Initial findings: We conducted a usability study of the online collaborative platform, in which one student in each team was responsible for screen sharing. Screen sharing was by far the most challenging aspect of using the platform—the platform has since been changed so that a facilitator trained in the use of the platform will share the screen instead. Also, during the usability study, we made an online shared whiteboard available. Students did not use the shared whiteboard when discussing these tasks. We concluded that while a shared whiteboard may be helpful if the tasks require it, in this instance, it added complexity to the platform and so we removed it. Based on our experience thus far in scoring responses from both individuals and teams, we have found generally high agreement between raters—however we have also found that high agreement is not necessarily an indication of a valid rubric.



  • Andrews-Todd, J., Graf, E. A., & Pinto, W. (2023, March). A collaborative problem-solving platform to measure understanding on a mathematics learning progression. Paper presented at the annual meeting of the National Council on Measurement in Education, Chicago, IL.
  • Graf, E. A. & Andrews-Todd, J. (2022, July). A usability study for collaborative problem solving in mathematics. Presentation for the We the People - Math Literacy For All & Algebra Project National Conference, virtual conference. https://youtu.be/bJH5emgvt20
  • Lizano, C., Lai, Y., & Young, C. (2022, July). Co-creating a learning progression with teachers that focuses on student strengths. Presentation for the We the People - Math Literacy For All & Algebra Project National Conference, virtual conference. https://youtu.be/4pJCAL1kYdo

We have also produced mathematics activities designed for use by teams of teachers or mentors that illustrate the levels of the LP/T.

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Learning Trajectories as a Complete Early Mathematics Intervention: Achieving Efficacies of Economies at Scale

PI: Douglas Clements
STEM Discipline: Mathematics
Grade Levels: preK

Project Description: The ULTIMATE (Understanding Learning Trajectories In Math: Advancing Teacher Education) research project supports teachers in deepening their understanding of how children learn mathematics and how to incorporate this understanding into their classrooms to help children develop math ideas and skills—joyfully. For a quarter of a century, Drs. Clements and Sarama have built a professional development tool through multiple research-and-development projects: Learning and Teaching with Learning Trajectories, or [LT]2. In ULTIMATE, the DU team is working with preschool teachers, blending high-quality in-person professional development with teachers' use of the [LT]2 tool and investigating the positive impacts in supporting teachers and on students' mathematics learning. [LT]2 allows teachers, caregivers, and parents to see and use learning trajectories for math as they view short video clips of good teaching and children working on math problems in a way that reveals their thinking. [LT]2 includes alignments with standards and assessments, and hands-on and built-in software for children. [LT]2 enables teachers to help children find the math in—and develop the math from—their everyday activities, including art, stories, puzzles, and games. ULTIMATE's learning trajectories approach is fundamentally asset-based, as all instruction builds upon children's level of thinking. We need new adaptations of existing empirically-validated interventions that deliver the greatest possible impact to the greatest number of students cost-effectively. The Learning Trajectories [LT]2 intervention represents the most promising opportunity thus far to demonstrate that impact can be achieved through economies of scale.

How does your project consider issues of equity when designing and/or assessing a learning trajectory?: As stated, ULTIMATE’s learning trajectories approach is fundamentally asset-based, as all instruction considers children’s level of thinking (groups and individuals). The approach does not "break down" mathematics into rote skills but builds up mathematics concepts, processes, and positive beliefs from what children can understand. Previous research clearly showed greater learning by children from marginalized communities using the learning trajectories approach.

How can we better prepare and support teachers to understand and effectively utilize learning progressions in their classrooms?: This is ULTIMATE’s goal: To support teachers(ULTIMATELY, birth to grade 2) in understanding and applying learning trajectories, whatever their situation and context, and to evaluate the effects of that support.

Initial findings: Findings from the first of two school district sites are being cleaned and analyzed, especially child outcome data (the second site started its first year of professional development this Fall). Initial findings from coaching logs, teacher surveys, teacher interviews, and observations of teachers showed good-to-excellent implementation and positive teacher reports. In addition, teachers grew in their support of students' conceptual understanding of mathematics concepts and processes and their ability to tailor instruction to meet students’ needs. For example, teachers at the beginning of the year viewed math as "right or wrong" and rarely probed for strategies. When children named a shape, the teacher's feedback was either, "Yes, that's right,' or "No." During the year, the teachers started asking children, "Why?' or "How do you know?" and filling in the student responses with additional information. When talking about patterns, some teachers moved from simply asking, "What comes next?" to having children name the pattern (e.g., ABB pattern) and generate other examples of that type of pattern. Finally, teachers reported that students demonstrated increased engagement and enjoyment in their math learning. 

Products: Reports and publications are planned over the next several years. An updated version of the Learning and Teaching with Learning Trajectories, or LT2, tool, is available to all (https://www.learningtrajectories.org/).


SPIRAL: Supporting Professional Inquiry and Re-Aligning Learning Through a Structured e-Portfolio System

PI: Jose-Felipe Martinez-Fernandez
STEM Discipline: Science
Grade Levels: 1-8

Project Description: The SPIRAL project is developing and testing a new model for vertical-team professional development, along with a set of electronic tools enabling collaboration to support instructional improvement aligned to the Next Generation Science Standards. Specifically, the project involves professional learning communities (PLCs) that comprise teachers across grades 1-8 to reflect the spiraling structure of Disciplinary Content Ideas (DCIs) in the NGSS. To address the need for scalable models of coherent professional reflection, we have built a digital portfolio and annotation platform (app and browser-based) that allows teachers to easily collect an array of artifacts (documents, image, and video) reflecting student evolving thinking and discourse, along with evidence of instruction and conceptual understanding; and to organize, share, and discuss this evidence in relation to DCI learning trajectories across grades.

Although the Framework for K-12 Science Education explicitly connects their work to learning progressions, there is limited guidance to teachers for organizing instruction to support these learning trajectories. The SPIRAL project is in part addressing learning trajectories in the following ways. Through professional learning experiences and the use of high-quality, NGSS-aligned storylines, we are supporting teachers’ increasingly complex understandings of learning trajectories across grade levels, and how classroom artifacts can support deeper, more coherent reflection around teaching and learning related to student thinking. The vertical PLCs are also helping us understand how teachers can collaborate to better understand levels of thinking in relation to learning trajectories for the topics of waves and water distribution.

How does your project consider issues of equity when designing and/or assessing a learning trajectory?: Participating teachers and students have been introduced to waves and water storylines designed to reflect how students develop understanding of the content standards, and the science practice of modeling. The storylines and accompanying professional learning for teachers focuses on a) anchoring events that draw place-based connections to relevant phenomena and issues directly affecting the students and their communities–for example water scarcity, and the disappearance of a local lake, and b) developing understanding and modeling skills in culturally responsive ways that allow all students to have their voices heard and their diverse perspectives and thinking used as assets.

What are the anticipated next steps in advancing research on this specific learning progression?: As part of our project, vertical teams of teachers focused on the same set of core disciplinary ideas have collected evidence of student thinking at multiple points throughout their grade-specific storylines. In particular, all teachers collected artifacts of students' initial, mid-point, and end-of-unit drawn conceptual models for the specific phenomenon in their storyline. As vertical teams they reviewed these artifacts to discuss the trajectory of thinking and learning across time points in the unit and across grades. We are analyzing a rich set of qualitative and quantitative data from different sources to identify changes in teacher thinking both about the core idea, and about progressions in student modeling abilities.

We also aim to create a publicly viewable repository of artifacts that shows progressions of learning across the micro (a unit) and macro (grade levels) with respect to particular ideas about water and waves.

Initial findings: Teacher reports after a second year of implementation point to significant value accrued from the PD interactions throughout the year. A salient theme in teacher interviews was surprise at the level of understanding demonstrated by students in earlier grades. Awareness of the DCIs and understandings targeted in earlier grades can be a powerful support to help teachers develop more challenging instruction and target more sophisticated understandings for students in their own grade. Interviews also suggest that discussions in vertical teams address issues beyond curriculum gaps and redundancies, to touch on aspects like teacher positioning, community, equitable instruction, and teaching identity.

Pre-post comparisons of both teacher and student surveys consistently point to greater levels of engagement with or exposure to NGSS practices in their classrooms. Moreover, teacher survey reports also reflect feelings of greater self-efficacy for NGSS instruction, and student survey reports reflect improved self-perceptions around science.


  • The SPIRAL app (Android and iOS) was created to capture, annotate, and share artifacts from the classroom.
  • A web-based platform that offers the same capabilities as the SPIRAL app, and additionally allows teachers to share, review, and discuss evidence of student thinking in their classroom.

Science of Teaching Math Logo

Usable Measures of Teacher Understanding: Exploring Diagnostic Models and Topic Analysis as Tools for Assessing Proportional Reasoning for Teaching

PI: Yasemin Copur-Gencturk
STEM Discipline: Mathematics
Grade Levels: Middle School Mathematics Teachers

Project Description: Scholars have leveraged the learning progression of students to improve student outcomes. Despite the fact that teachers continuously participate in professional development activities to enhance their knowledge and skills, relatively limited attention has been given to teachers’ learning progression. The purpose of this project is to develop an assessment to gather information regarding what teachers’ learning progression looks like as a consequence of their participation in mathematics-focused professional development programs. Our approach is novel in that we capture the nuances of teachers’ understanding through open-text responses, and we create a statistical model that provides insights into the development of key elements of teachers’ content and pedagogical content knowledge. This approach has potential to identify the trends in teachers’ knowledge development and to provide guidance to teacher educators on how effectively their programs improve certain elements of teachers’ knowledge. In addition, identifying teachers’ progression has the potential to shed light on how teachers with different learning progressions contribute to their students’ learning trajectories.

What are the anticipated next steps in advancing research on this specific learning progression?: Advances in artificial intelligence make it possible to develop measures that consist of open-text responses and to analyze the trends in teachers’ learning more effectively.

Initial findings: We have identified the ways teachers reason in proportional situations and have found that topic modeling can produce results similar to a separate, qualitative analysis of the teachers’ responses.  https://link.springer.com/article/10.1007/s10857-021-09529-w

Products: Project website

Additional Projects

We invite you to explore a sample of the other recently awarded and active work that focuses on learning progressions and trajectories in the DRK-12 portfolio.

Related Resources


Andrews-Todd, J., & Forsyth, C. M. (2020). Exploring social and cognitive dimensions of collaborative problem solving in an open online simulation-based task. Computers in Human Behavior, 104, 105759. https://doi.org/10.1016/j.chb.2018.10.025

Borko, H., Koellner, K., Jacobs, J., & Seago, N. (2011). Using video representations of teaching in practice-based professional development programs. ZDM - International Journal on Mathematics Education43(1), 175–187. https://doi.org/10.1007/s11858-010-0302-5

Confrey, J. (2019). A Synthesis of Research on Learning Trajectories/Progressions in Mathematics. Commissioned for the OECD 2030 Learning Framework, by OECD Mathematics Curriculum Document Analysis Project Workshop. Access: http://www.oecd.org/education/2030-project/about/documents/A_Synthesis_of_Research_on_Learning_Trajectories_Progressions_in_Mathematics.pdf

Eames, C. L., Graf, E. A., van Rijn, P. W., Budzban, G., & Voepel, T. (2021). The finite-to-finite strand of a learning progression for the concept of function: A research synthesis and cognitive analysis. The Journal of Mathematical Behavior, 62, 100864. https://doi.org/10.1016/j.jmathb.2021.100864

Graf, E. A., van Rijn, P. W., & Eames, C. L. (2021). A cycle for validating a learning progression illustrated with an example from the concept of function. The Journal of Mathematical Behavior, 62, 100836. https://doi.org/10.1016/j.jmathb.2020.100836

Horn, I., & Garner, B. (2022). Teacher learning of ambitious and equitable mathematics instruction: A sociocultural approach (1st ed.). Routledge. https://doi.org/10.4324/9781003182214

Marshall, S. A. (2020). Responsive, locally-relevant coaching: Supporting STEM teachers' learning of justice-oriented pedagogies. [Doctoral dissertation, Vanderbilt University]

Marshall, S. A. (2022). Using problem (re)framing and teachers’ pedagogical responsibility to facilitate teacher learning opportunities. In C. Chinn, E. Tan, C. Chan, & Y. Kali (Eds.), Proceedings of the 16th International Conference of the Learning Sciences-ICLS 2022 (pp. 409–416). International Society of the Learning Sciences.

Michaels, S., & O’Connor, C. (2012). Talk science primer. Cambridge, MA: TERC. Access: https://inquiryproject.terc.edu/shared/pd/TalkScience_Primer.pdf

Moses, R., & Cobb, C. E. (2001). Radical equations: Civil rights from Mississippi to the Algebra Project. Boston, MA: Beacon Press.

Moses, R., Kamii, M., Swap, S. M., & Howard, J. (1989). The algebra project: Organizing in the spirit of Ella. Harvard Educational Review, 59(4), 423–444. https://doi.org/10.17763/haer.59.4.27402485mqv20582

Moses, R., West, M. M., & Davis, F. E. (2009). Culturally responsive mathematics education in the Algebra Project. In B. Greer, S. Mukhopadhyay, A. B. Powell, & S. Nelson-Barber (Eds.), Culturally responsive mathematics education (pp. 239–256). New York: Routledge.

Yeh, C. (2021). Responsive and relevant to whom? Mathematics Teacher: Learning and Teaching PK-12114(1), 83–84. https://doi.org/10.5951/MTLT.2020.0083