Quantitative

Moving Beyond Pedagogy: Developing Elementary Teachers' Adaptive Expertise in Using the Epistemic Complexity of Science

The Next Generation Science Standards (NGSS) emphasize the integration of scientific knowledge and the practices of science, a recognition that science classrooms are complex learning environments. Meeting this expectation requires teachers to move beyond traditional routines of practice to become adaptive experts who can adjust their teaching to maximize learning in varied classroom situations.

Lead Organization(s): 
Award Number: 
1812576
Funding Period: 
Tue, 05/15/2018 to Sat, 04/30/2022
Full Description: 

The Next Generation Science Standards (NGSS) emphasize the integration of scientific knowledge and the practices of science, a recognition that science classrooms are complex learning environments. Meeting this expectation requires teachers to move beyond traditional routines of practice to become adaptive experts who can adjust their teaching to maximize learning in varied classroom situations. A teacher who has adaptive expertise is defined as someone who can self-assess and strategically adjust decision-making before, during and after teaching episodes. To become adaptive experts, teachers must understand the foundational ways that scientific knowledge is advanced and develop knowledge of, and practices related to, using argument, language, and dialogical environments--individually and collectively--as tools for learning science. To effectively use these tools requires teachers to shift from viewing science teaching as the transfer or replication of knowledge through routines of practices to one in which students are participants in a more cognitively based approach to learning. How teachers develop adaptive expertise for NGSS-aligned learning environments is still little understood. This project will examine the complex nature of the relationship between these learning tools and teacher orientation that enables teachers to develop adaptive expertise over the course of a multi-year professional development program.

The project will work with 150 Grade 3-5 teachers in Iowa and Alabama to implement a three-year professional development program to assist teachers develop adaptive expertise. Through implementation of an argument-based inquiry approach focused on development of adaptiveness, teachers will be supported as they shift their expertise from routine to adaptiveness. Project data will include teachers' implementation of the approach, their understanding of science argument, and their shifting epistemic orientation. The project will examine selected case studies of teachers to better understand the variations in development of adaptive expertise. The project outcome will be a model of adaptive expertise that can be used by in-service and pre-service educators to advance teacher practices towards adaptive expertise. The aim is to design ways to transfer adaptive expertise to students in STEM. The mixed-method project will integrate analyses with a focus on understanding complexity, using large-scale quantitative data.

CAREER: Supporting Elementary Science Teaching and Learning by Integrating Uncertainty Into Classroom Science Investigations

The goal of this study is to improve elementary science teaching and learning by developing, testing, and refining a framework and set of tools for strategically incorporating forms of uncertainty central to scientists' sense-making into students' empirical learning.

Lead Organization(s): 
Award Number: 
1749324
Funding Period: 
Fri, 06/01/2018 to Wed, 05/31/2023
Full Description: 

The goal of this study will be to improve elementary science teaching and learning by developing, testing, and refining a framework and set of tools for strategically incorporating forms of uncertainty central to scientists' sense-making into students' empirical learning. The framework will rest on the notion that productive uncertainty should be carefully built into students' empirical learning experiences in order to support their engagement in scientific practices and understanding of disciplinary ideas. To re-conceptualize the role of empirical investigations, the study will focus on the transitions between the experiences and processes students seek to understand, classroom investigations, evidence, and explanatory models as opportunities for sense-making, and how uncertainty can be built into these transitions. The project's underlying assumption is that carefully implementing these forms of uncertainty will help curriculum developers and teachers avoid the oversimplified investigations that are prevalent in K-8 classrooms that stand in stark contrast to authentic science learning and the recommendations of the Framework for K-12 Science Education (National Research Council, 2012). Accordingly, the project will seek to develop curriculum design guidelines, teacher tools, professional development supports, and four elaborated investigations, including sets of lessons, videos, and assessments that embed productive uncertainty for second and fifth graders and designed for use with linguistically, culturally, and socio-economically diverse students.

The hypothesis of this work is that if specific forms of scientific uncertainty are carefully selected, and if teachers can implement these forms of uncertainty, elementary students will have more robust opportunities to develop disciplinary practices and ideas in ways consistent with the Next Generation Science Standards (NGSS) (Lead States, 2013). Employing Design-Based Research, the three research questions will be: (1) What opportunities for sense-making do elementary school empirical investigations afford where we might strategically build uncertainty?; (2) How can we design learning environments where uncertainty in empirical investigations supports opportunities for learning?; and (3) In classrooms with sustained opportunities to engage with uncertainty in empirical investigations, what progress do students make in content understandings and the practices of argumentation, explanation, and investigation? The work will consist of three design cycles: Design Cycle 1 will involve two small groups of six teachers in adapting their curricula to incorporate uncertainty, then describe how students engage around uncertainty in empirical investigations. Design Cycle 2 will involve the same small groups in implementing and refining task structures, tools, and teacher instructional strategies. In Design Cycle 3, teachers and researchers will further refine lesson materials, assessments, and supports. The project will partner with one school district and engage in design research with groups of teachers to develop: (1) a research-based description, with exemplars of opportunities for student sense-making within empirical investigations at both early and upper elementary grades; (2) a set of design principles and tools that allow teachers to elicit and capitalize on sense-making about uncertainty in investigations; and (3) four elementary investigations elaborated to incorporate and exemplify the first two products above. These materials will be disseminated through a website, and established networks for supporting implementation of the NGSS. An advisory board will oversee project progress and conduct both formative and summative evaluation.

Measuring Early Mathematical Reasoning Skills: Developing Tests of Numeric Relational Reasoning and Spatial Reasoning

The primary aim of this study is to develop mathematics screening assessment tools for Grades K-2 over the course of four years that measure students' abilities in numeric relational reasoning and spatial reasoning. The team of researchers will develop Measures of Mathematical Reasoning Skills system, which will contain Tests of Numeric Relational Reasoning (T-NRR) and Tests of Spatial Reasoning (T-SR).

Award Number: 
1721100
Funding Period: 
Fri, 09/15/2017 to Tue, 08/31/2021
Full Description: 

Numeric relational reasoning and spatial reasoning are critical to success in later mathematics coursework, including Algebra 1, a gatekeeper to success at the post-secondary level, and success in additional STEM domains, such as chemistry, geology, biology, and engineering. Given the importance of these skills for later success, it is imperative that there are high-quality screening tools available to identify students at-risk for difficulty in these areas. The primary aim of this study is to develop mathematics screening assessment tools for Grades K-2 over the course of four years that measure students' abilities in numeric relational reasoning and spatial reasoning. The team of researchers will develop Measures of Mathematical Reasoning Skills system, which will contain Tests of Numeric Relational Reasoning (T-NRR) and Tests of Spatial Reasoning (T-SR). The measures will be intended for use by teachers and school systems to screen students to determine who is at-risk for difficulty in early mathematics, including students with disabilities. The measures will help provide important information about the intensity of support that may be needed for a given student. Three forms per grade level will be developed for both the T-NRR and T-SR with accompanying validity and reliability evidence collected. 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 development of the T-NRR and T-SR measures will follow an iterative process across five phases. The phases include (1) refining the construct; (2) developing test specifications and item models; (3) developing items; (4) field testing the items; and (5) conducting validity studies. The evidence collected and evaluated during each phase will contribute to the overall evaluation of the reliability of the measures and the validity of the interpretations made using the measures. Item models, test specifications, and item development will be continuously evaluated and refined based on data from cognitive interviews, field tests, and reviews by mathematics educators, teachers of struggling students, teachers of culturally and linguistically diverse populations, and a Technical Advisory Board. In the final phase of development of the T-NRR and T-SR, reliability of the results will be estimated and multiple sources of validity evidence will be collected to examine the concurrent and predictive relation with other criterion measures, classification accuracy, and sensitivity to growth. Approximately 4,500 students in Grades K-2 will be involved in all phases of the research including field tests and cognitive interviews. Data will be analyzed using a two-parameter IRT model to ensure item and test form comparability.


Project Videos

2020 STEM for All Video Showcase

Title: Measuring Early Mathematical Reasoning Skills

Presenter(s): Leanne Ketterlin Geller


A Partnership to Adapt, Implement and Study a Professional Learning Model and Build District Capacity to Improve Science Instruction and Student Understanding (Collaborative Research: Borko)

This project will work in partnership with the Santa Clara Unified School District (SCUSD) to adapt a previously designed Professional Learning (PL) model based on the District's objectives and constraints to build the capacity of teacher leaders and a program coordinator to implement the adapted PL program. The project is examining the sustainability and scalability of a PL model that supports the development of teachers' pedagogical content knowledge and instructional practices.

Lead Organization(s): 
Award Number: 
1720930
Funding Period: 
Sun, 10/01/2017 to Thu, 09/30/2021
Full Description: 

The Lawrence Hall of Science (the Hall) and Stanford University teams have previously developed and tested the efficacy of a program of Professional Learning (PL) which is focused on improving teachers' ability to support students' ability to engage in scientific argumentation. Key components of the PL model include a week-long summer institute and follow-up sessions during the academic year that incorporate additional pedagogical input, video reflection, and planning time. In this project, the Hall and Stanford are working in partnership with the Santa Clara Unified School District (SCUSD) to adapt the PL model based on the District's objectives and constraints, to build the capacity of teacher leaders and a program coordinator to implement the adapted PL program. This will enable the District to continue to adapt and implement the program independently at the conclusion of the project. Concurrently, the project is studying the adaptability of the PL model and the effectiveness of its implementation, and is developing guidelines and tools for other districts to use in adapting and implementing the PL model in their local contexts. Thus, this project is contributing knowledge about how to build capacity in districts to lead professional learning in science that addresses the new teaching and learning standards and is responsive to the needs of their local context.

The project is examining the sustainability and scalability of a PL model that supports the development of teachers' pedagogical content knowledge and instructional practices, with a particular focus on engaging students in argument from evidence. Results from the Hall and Stanford's previous research project indicate that the PL model is effective at significantly improving teachers' and students' classroom discourse practices. These findings suggest that a version of the model, adapted to the context and needs of a different school district, has the potential to improve the teaching of science to meet the demands of the current vision of science education. Using a Design-Based Implementation Research approach, this project is (i) working with SCUSD to adapt the PL model; (ii) preparing a district project coordinator and cadre of local teacher leaders (TLs) to implement and further adapt the model; and (iii) studying the adaptation and implementation of the model. The outcomes will be: a) a scalable PL model that can be continually adapted to the objectives and constraints of a district; b) a set of activities and resources for the district to prepare and support the science teacher leaders who will implement the adapted PL program internally with other teachers; and c) knowledge about the adaptations and resources needed for the PL model to be implemented independently by other school districts. The team also is researching the impact of the program on classroom practices and student learning.


Project Videos

2020 STEM for All Video Showcase

Title: Accomplishments and Struggles in a 3-Way RPP

Presenter(s): Emily Weiss, Hilda Borko, Coralie Delhaye, Jonathan Osborne, Emily Reigh, Tricia Ringel, Craig Strang, & Krista Woodward

2019 STEM for All Video Showcase

Title: Building District Leadership in Scientific Argumentation

Presenter(s): Coralie Delhaye, Emily Reigh, & Emily Weiss

2018 STEM for All Video Showcase


Project MAPLE: Makerspaces Promoting Learning and Engagement

The project plans to develop and study a series of metacognitive strategies that support learning and engagement for struggling middle school students during makerspace experiences. The study will focus narrowly on establishing a foundational understanding of how to ameliorate barriers to engaging in design learning through the use of metacognitive strategies.

Award Number: 
1721236
Funding Period: 
Fri, 09/01/2017 to Sat, 08/31/2019
Full Description: 

The project plans to develop and study a series of metacognitive strategies that support learning and engagement for struggling middle school students during makerspace experiences. The makerspace movement has gained recognition and momentum, which has resulted in many schools integrating makerspace technologies and related curricular practices into the classroom. The study will focus narrowly on establishing a foundational understanding of how to ameliorate barriers to engaging in design learning through the use of metacognitive strategies. The project plans to translate and apply research on the use of metacognitive strategies in supporting struggling learners to develop approaches that teachers can implement to increase opportunities for students who are the most difficult to reach academically. Project strategies, curricula, and other resources will be disseminated through existing outreach websites, research briefs, peer-reviewed publications for researchers and practitioners, and a webinar for those interested in middle-school makerspaces for diverse learners.

The research will address the paucity of studies to inform practitioners about what pedagogical supports help struggling learners engage in these makerspace experiences. The project will focus on two populations of struggling learners in middle schools, students with learning disabilities, and students at risk for academic failure. The rationale for focusing on metacognition within makerspace activities comes from the literature on students with learning disabilities and other struggling learners that suggests that they have difficulty with metacognitive thinking. Multiple instruments will be used to measure metacognitive processes found to be pertinent within the research process. The project will tentatively focus on persistence (attitudes about making), iteration (productive struggle) and intentionality (plan with incremental steps). The work will result in an evidence base around new instructional practices for middle school students who are struggling learners so that they can experience more success during maker learning experiences.

Engaging Students in Scientific Practices: Evaluating Evidence and Explanation in Secondary Earth and Space Science

This project will develop, implement, test, and revise instructional approaches and materials for high school students that focus on the links between scientific evidence and alternative explanations of phenomena relating to Earth and space education. Students will learn to construct diagrams showing the links between explanatory models of natural phenomena and lines of evidence, and then evaluate the plausibility of various alternative explanations for events.

Award Number: 
2027376
Funding Period: 
Fri, 09/01/2017 to Tue, 08/31/2021
Full Description: 

This project will develop, implement, test, and revise instructional approaches and materials for high school students that focus on the links between scientific evidence and alternative explanations of phenomena relating to Earth and space education. Students will examine alternative explanations for natural phenomena associated with extreme weather events, freshwater resource availability, and related topics in learning how to evaluate scientifically valid lines of evidence and explanation. Students will learn to construct diagrams showing the links between explanatory models of natural phenomena and lines of evidence, and then evaluate the plausibility of various alternative explanations for events. It is expected that engagement in these activities will help students gain proficiency in model-based reasoning, critical thinking, planning and analyzing scientifically valid investigations, constructing plausible explanations, engaging in collaborative argumentation, and critically evaluating scientific information.

This 4-year Design and Development project will examine use of Model-Evidence Link (MEL) diagrams that are intended to help students cognitively construct mental scaffolds that assist their engagement in the practices of critical evaluation, plausibility appraisal, and knowledge construction related to science topics that are considered by some as controversial. Prior research has demonstrated the potential educational outcomes of using MEL diagrams, but this project will extend the previous work by examining an approach where students construct their own MEL diagrams (build-a-MELs, or baMELs). The project will examine the use of both pre-constructed MELs and baMELs for effectiveness in promoting student engagement in scientific reasoning and practices. The project will employ design-based research methodologies in pursuing answers to three research questions: (1) Do baMEL activities tested in multiple high school classroom settings promote critical evaluation, plausibility reappraisal, and  scientifically accurate knowledge construction about controversial Earth and space science topics? (2) How do these additional baMELs differ from pre-constructed MELs in promoting critical evaluation, plausibility reappraisal, and knowledge construction? And (3) To what extent does repeated use of both pre-constructed MELs and baMELs result in student engagement of scientific practices (i.e., asking critical questions, using model-based reasoning, planning and analyzing scientifically valid investigations, constructing plausible explanations, engaging in collaborative argumentation, and critically evaluating scientific information)? The project will engage high school students taking Earth and space classes in selected schools of Georgia, New Jersey, and within Philadelphia. Teacher professional development opportunities associated with the project will include summer institutes, classroom supports, and mentoring sessions.

This project was previously funded under award #1721041.

Project Accelerate: University-High School AP Physics Partnerships

Project Accelerate blends the supportive structures of a student's home school, a rigorous online course designed specifically with the needs of under-served populations in mind, and hands-on laboratory experiences, to make AP Physics accessible to under-served students. The project could potentially lead to the success of motivated but under-served students who attend schools where the opportunity to engage in a rigorous STEM curriculum is not available.

Lead Organization(s): 
Award Number: 
1720914
Funding Period: 
Tue, 08/01/2017 to Fri, 07/31/2020
Full Description: 

Project Accelerate brings AP Physics 1 and, eventually, AP Physics 2 to students attending schools that do not offer AP Physics. The project will enable 249 students (mostly under-served, i.e., economically disadvantaged, ethnic minorities and racial minorities) to enroll in AP Physics - the students would otherwise not have access. These students either prepare for the AP Physics 1 exam by completing a highly interactive, conceptually rich, rigorous online course, complete with virtual lab experiments, or participate in an accredited AP course that also includes weekly hands-on labs. In this project, the model will be tested and perfected with more students and expanded to AP Physics 2. Further, model replication will be tested at an additional site, beyond the two pilot sites. In the first pilot year in Massachusetts at Boston University, results indicated that students fully engaged in Project Accelerate are (1) at least as well prepared as peer groups in traditional classrooms to succeed on the AP Physics 1 exam and (2) more inclined to engage in additional STEM programs and to pursue STEM fields and programs than they were prior to participating. In the second year of the pilot study, Project Accelerate doubled in size and expanded in partnership with West Virginia University. From lessons learned in the pilot years, key changes are being made, which are expected to increase success. Project Accelerate provides a potential solution to a significant national problem of too few under-served young people having access to high quality physics education, often resulting in these students being ill prepared to enter STEM careers and programs in college. Project Accelerate is a scalable model to empower these students to achieve STEM success, replicable at sites across the country (not only in physics, but potentially across fourteen AP subjects). The project could potentially lead to the success of tens of thousands of motivated but under-served students who attend schools where the opportunity to engage in a rigorous STEM curriculum is not available.

Project Accelerate blends the supportive structures of a student's home school, a private online course designed specifically with the needs of under-served populations in mind, and hands-on laboratory experiences, to make AP Physics accessible to under-served students. The goals of the project are: 1) have an additional 249 students, over three years, complete the College Board-accredited AP Physics 1 course or the AP Physics 1 Preparatory course; 2) add an additional replication site, with a total of three universities participating by the end of the project; 3) develop formal protocols so Project Accelerate can be replicated easily and with fidelity at sites across the nation; 4) develop formal protocols so the project can be self-sustaining at a reasonable cost (about $500 per student participant); 5) build an AP Physics 2 course, giving students who come through AP Physics 1 a second year of rigorous experience to help further prepare them for college and career success; 6) create additional rich interactive content, such as simulations and video-based experiments, to add to what is already in the AP Physics 1 prep course and to build the AP Physics 2 prep course - the key is to actively engage students with the material and include scaffolding to support the targeted population; 7) carry out qualitative and quantitative education research, identifying features of the program that work for the target population, as well as identifying areas for improvement. This project will support the growing body of research on the effectiveness of online and blended (combining online and in-person components) courses, and investigate the use of such courses with under-represented high school students.

The Mathematical Knowledge for Teaching Measures: Refreshing the Item Pool

This project proposes an assessment study that focuses on improving existing measures of teachers' Mathematical Knowledge for Teaching (MKT). The research team will update existing measures, adding new items and aligning the instrument to new standards in school mathematics.

Lead Organization(s): 
Award Number: 
1620914
Funding Period: 
Thu, 12/01/2016 to Sat, 11/30/2019
Full Description: 

This project proposes an assessment study that focuses on improving existing measures of teachers' Mathematical Knowledge for Teaching (MKT). The research team will update existing measures, adding new items and aligning the instrument to new standards in school mathematics. In addition, the team will update the delivery system for the assessment to Qualtrics, a more flexible online system.

The research team will build an updated measure of teachers' Mathematical Knowledge for Teaching (MKT). Project researchers will conduct item writing camps, develop new items, cognitively pilot and revise items, and factor analyze items. The researchers will also determine item constructs and calibrate items (and constructs) through an innovative application of Item Response Theory (IRT) employing a variant of the standard 2-parameter IRT model. Finally, the team will oversee the transition of the Teacher Knowledge Assessment System to the Qualtrics data collection environment to allow for more flexible item specification.

Science and Engineering Education for Infrastructure Transformation

This project focuses on the research and develop an engineering education technology and pedagogy that will support project-based learning of science, engineering, and computation concepts and skills underlying the strategically important "smart" and "green" aspects of the infrastructure. The project will develop transformative technologies and curriculum materials to turn the campus of a high school or a geographical information system such as Google Maps into an engineering laboratory with virtually unlimited opportunities for learning and exploration.

Lead Organization(s): 
Partner Organization(s): 
Award Number: 
1721054
Funding Period: 
Sun, 10/01/2017 to Thu, 09/30/2021
Full Description: 

The Concord Consortium in collaboration with Purdue University will research and develop an engineering education technology and pedagogy that will support project-based learning of science, engineering, and computation concepts and skills underlying the strategically important "smart" and "green" aspects of the infrastructure. This project will develop transformative technologies and curriculum materials to turn the campus of a high school or a geographical information system such as Google Maps into an engineering laboratory with virtually unlimited opportunities for learning and exploration. The project will deliver two innovations: 1) The Smart High School is an engineering platform for designing Internet of Things systems for managing the resources, space, and processes of a school based on real-time analysis of data collected by various sensors deployed by students on campus; and 2) the Virtual Solar World is a computational modeling platform for students to design, deploy, and connect virtual solar power solutions for their homes, schools, and regions. Six standards-aligned curriculum units based on these technologies will be developed to guide student learning and support educational research. Approximately 2,000 students from rural, suburban, and urban high schools in Indiana, Massachusetts, New Hampshire, and Ohio will participate in this research. project products and findings through the Internet, conferences, publications, and partner networks.

The research is designed to identify technology-enhanced instructional strategies that can simultaneously foster the growth of skills and self-efficacy in scientific reasoning, design thinking, and computational thinking, all of which are needed to build the future infrastructure. The focus on infrastructure transformation is aligned with NSF's vision of smart and connected communities. Although this project will use the context of smart and green infrastructure to engage students to solve real-world problems, the skills of scientific reasoning, design thinking, and computational thinking that they will acquire through meeting the challenges of this project can be transferrable to other topics and fields. Using a design-based research approach, a rich set of formative and summative data will be collected from these students for probing into three research questions: 1) To what extent does the integrated learning model help students develop and connect scientific reasoning, design thinking, and computational thinking skills?; 2) To what extent is students' interest in cognate careers affected by the authenticity of engineering design challenges?; and 3) How do the variations in the solutions to overcome the cognitive and practical difficulties of real-world problems impact learning outcomes and career interest? The data sources include pre/post-tests, process data, self-reports, observations, surveys, interviews, and participant information.

Examining Relationships Between Flipped Instruction and Students' Learning of Mathematics

This study can provide a basis for design research focused on developing effective materials and programs for flipped instruction in secondary mathematics, which is already occurring at an increasing rate, but it is not yet informed by empirical evidence. This project will result in a framework for flipped instruction robust enough to be useful at a variety of grade levels and contexts. The framework will provide a better understanding of the relationships among various implementations of flipped instruction and student learning.

Lead Organization(s): 
Award Number: 
1721025
Funding Period: 
Tue, 08/01/2017 to Fri, 07/31/2020
Full Description: 

Instead of presenting new material in class and then assigning problems to be completed outside of class, flipped instruction involves students watching videos or reading new material outside of class and then completing their "homework" in class. Teachers' implementation of flipped instruction has increased dramatically in recent years, with more than two-thirds of teachers now reporting flipping a lesson, if not an entire course. Although popular media and philanthropic organizations have given a great deal of attention and financial support to flipped instruction, little is known about how teachers implement it and what benefits and drawbacks flipped instruction has in contrast with non-flipped instruction. This study can provide a basis for design research focused on developing effective materials and programs for flipped instruction in secondary mathematics. This design and development is already occurring at an increasing rate, but it is not yet informed by empirical evidence. This project will result in a framework for flipped instruction robust enough to be useful at a variety of grade levels and contexts. The framework will provide a better understanding of the relationships among various implementations of flipped instruction and student learning. These findings can inform teacher educators in better aligning their instruction to instructional formats that correlate with increased student learning outcomes.

Using mixed-methods techniques, the study will look at the nature of the activities and interactions occurring in mathematics classrooms and assess their quality so that the researchers may distinguish high-quality from low-quality univocal discourse, high-quality from low-quality dialogic discourse, and high cognitive demand from low cognitive demand tasks. Working in 40 algebra classrooms -- 20 implementing some form of flipped instruction and 20 serving as a non-flipped basis for comparison -- the project will address the following research questions using a correlational design and multilevel modeling techniques: RQ1. What are salient factors entailed in flipped instruction in secondary algebra? RQ2. What associations, if any, exist among factors entailed in teachers' implementation of flipped algebra instruction and students' learning of algebra as measured on a state-mandated end-of-course assessment and on a concept-of-variable inventory?

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