Content-Referenced Growth

An Overview of Content-Referenced Growth

CADRE has been involved in a Research-Practice Partnership with assessment specialists from the organization Curriculum Associates on a project that is exploring innovations in the presentation and interpretation of student growth from large-scale educational assessments.

CADRE’s central innovation in this space is what we have come to describe as taking a “content-referenced growth reporting” approach, as distinct from the two more predominant approach to thinking about student growth (norm-referenced and criterion-referenced growth). The idea is to shift attention from growth interpretations that emphasize “points earned” to growth interpretations that focus on conceptual changes in student thinking, changes that can be characterized in concrete terms by the content of the assessment in the form of exemplar items.

The embodiment of this idea is found in a prototype we have developed for an interactive digital score reporting interface. CRG reporting is intended to help teachers understand the changes in thinking that occur as students advance in their learning of different skills and standards across content areas. The goal is for teachers to identify common ways that students think about problems, and to connect changes in this thinking to the growth a student has made in assessment scores.

The reporting interface is designed to accomplish this by giving users the opportunity to visualize and interpret student growth across multiple assessment occasions, using a score scale whose interpretation is facilitated by qualitatively distinct reference locations. Importantly, we also want the reporting interface to be informed by cognitive principles for the effective representation of quantitative data, and by the intended answers—as well as new questions—that the interface should provide and elicit about student growth.

Read the paper: Visualizing and Reporting Content-Referenced Growth on a Learning Progression (2025)

This document introduces a framework forcontent-referenced growth (CRG) reportingin educational assessments, designed to provide teachers withmeaningful insights into student progress. The authors explain that traditional score reports often lack qualitative depth, focusing merely onnumerical growth targetsrather thanconceptual understanding. The proposed CRG approach integrateslearning progressions, which describe increasingly sophisticated ways of thinking about a topic, withitem mappingand avertically scaled assessment like the i-Ready Diagnostic, to visually represent student growth along these progressions.

A pilot study with teachers using a prototype for understanding fractions revealed that the interface facilitatedconnections between scale scores and learning progression levels, although some teachers still relied on prior normative interpretations. The article concludes by discussing thevalidity of CRG interpretationsand suggesting future research to enhance its effectiveness and widespread adoption in supportingteacher professional learningandimproved student outcomes.

An example of CADRE’s digital CRG reporting prototype specific to student understanding of fractions can be accessed at (fractions).

Two of the major inspirations for the CRG approach come from item mapping, as instantiated in the , and the research literature on learning progressions (e.g., Clements & Sarama, 2004; Alonzo & Gotwals, 2012;). Other related explications of the ideas behind CRG can be found in Briggs & Peck (2015), Briggs (2019) and Wilson (2023).

Figure 1. Theory of Action Behind CRG Reporting

Figure 1. Theory of Action Behind CRG Reporting

A high-level theory of action for CRG reporting is outlined in Figure 1. The top of the diagram starts with the end goal which is to facilitate improved learning outcomes for students. This is accomplished through teachers adjusting their instruction based upon the inferences about growth and learning they can make from using the CRG reporting interface. There are two secondary goals of the prototype; 1) to contribute to teachers’ professional learning, and 2) to improve attitudes about the usefulness of assessment. We believe that as teachers attempt to connect student growth to changes along an underlying learning progression, they will also be engaging in a process of professional learning that deepens their content area expertise. Similarly, as teachers learn—with support—to interact with the reporting interface, we hope they develop an improved relationship with assessment and data use. In the Figure 1 diagram, the foundation for the CRG reporting interface depicted by four building blocks: a domain-specific learning progression, targeted test design, a scale established using item response theory (i.e., the Rasch model), and items mapped from the scale to levels of the learning progression.

To date we have developed prototypes for a CRG reporting interface specific to learning progressions for the core concepts of fractions, functions, ratio and spatial measurement in mathematics as depicted in Figure 2. We have also developed a CRG prototype for phonics knowledge as a foundational skill for reading.

Figure 2. CRG Reporting Prototypes and Learning Progressions in Mathematics

Figure 2. CRG Reporting Prototypes and Learning Progressions in Mathematics

The most complete explication of CRG reporting can be found on a 2025 publication in the journal Educational Assessment, "."

Reports on learning progression development and validation in the domains of mathematics and literacy can be found in Wellberg, Briggs & Student (2023) and Cox & Briggs (2023). We have engaged in two rounds of interviews with teachers in 2022 and 2023 as they have interacted with the CRG protoypes for fractions and phonics. For reports on these interviews and lessons learned, see Briggs et al. (2023) and Buchbinder et al. (2023).

The CADRE PI for the CRG research project is Derek Briggs. Major contributions have come from current and former CADRE graduate student researchers (in alphabetical order) Nicolas Buchbinder, Olivia Cox, Kyla McClure, Sandy Student, Sarah Wellberg and Erik Whitfield.

References

Briggs, D. C. & Peck, F. A. (2015). Using learning progressions to designvertical scales that support coherent inferences about student growth. Measurement: Interdisciplinary Research & Perspectives, 13, 75-99.

Briggs, D. C. (2019). Interpreting and visualizing the unit of measurement in the Rasch Model. Measurement, 46 (2019) 961–971.

Clements, D. H., & Sarama, J. (2004). Learning Trajectories in Mathematics Education. Mathematical Thinking and Learning, 6(2), 81–89. https://doi.org/10.1207/s15327833mtl0602_1

Alonzo, A. & Gotwals, A. (2012). Learning progressions in science: current challenges and future directions(2012). SensePublishers.

Wilson, M. (2023). Constructing measures : an item response modeling approach. Routledge.

Content-Referenced Growth FAQ

Content-Referenced Growth (CRG) reporting is an innovative framework for visualizing and communicating large-scale assessment results, specifically designed to help teachers make content-referenced interpretations of student growth. Unlike traditional assessment reporting, which often focuses on purely normative metrics (e.g., percentiles) or criterion-referenced targets ("growth to standard" based on numerical scale scores), CRG emphasizes providing qualitative, instructionally meaningful insights into what a student has learned and how their understanding has evolved over time.

Traditional reports might tell a teacher that a student grew by 20 scale score points, but CRG aims to translate that numerical growth into a concrete understanding of how the student's conceptualization of a subject has become more sophisticated. For example, instead of just a score increase, a CRG report could illustrate that a student progressed from understanding fractions as "part-whole" relationships to understanding them as "measurements" on a number line. This approach is motivated by the goal of supporting teachers in making conceptual adjustments to their instruction, rather than just procedural ones, by connecting observed growth directly to specific stages of a learning progression.

The CRG framework is built upon four interconnected core elements that, when combined, offer a novel approach to score reporting:

  1. Item Mapping: This involves using the qualitative distinctions among assessment items to help interpret numerical distances along a score scale. The CRG approach emphasizes how these distances correspond to instructionally meaningful differences in content understanding, by identifying classes of items that can be ordered by difficulty and linked to the time needed for instruction.
  2. Learning Progressions: Also known as progress maps or learning trajectories, these are research-based theories that describe the successively more sophisticated ways students think about a particular topic over time. These progressions typically include discrete levels or "waypoints" that represent significant intermediate steps in understanding. A learning progression provides the theoretical rationale for the ordering of item difficulties and forms the basis for content-referenced interpretations of growth.
  3. Test Design: CRG requires tests to be purposefully designed with overlapping item content across different test occasions (e.g., grades). This "spiraled" content design, similar to common item nonequivalent group designs, allows for distinguishing differences in student proficiency from the difficulty of items and ensures a common scale can be established.
  4. Vertical Scale Calibration (using the Rasch Model): This element involves placing occasion-specific tests onto a common vertical scale. The Rasch Model is preferred because, when it fits the data adequately, it allows for invariant comparisons between persons regardless of the specific items used to define a reference interval. This ensures that the interpretation of distances along the scale remains consistent and meaningful.

Consider the example of how people come to understand fractions. The CRG approach uses a specific learning progression (LP) to help students, teachers, and parents interpret growth in this domain. This LP draws upon established theories of fractions and is structured into four distinct levels, designed to represent increasing sophistication in conceptual understanding:

  1. Part-Whole: At this foundational level, students understand fractions primarily as a specified number of parts out of a total number of parts, often through visual relationships (e.g., identifying shaded portions of a shape).
  2. Quotient (renamed "Fair Shares" in the prototype): Students at this level grasp that fractions represent the division of a whole into equal parts. They understand unit fractions and how to "share" a whole among groups.
  3. Measurement (renamed "Number Line" in the prototype): This level signifies students' ability to understand fractions as magnitudes that can be located on a number line. They can order fractional values and understand equivalent fractions, which is crucial for operations like adding fractions with unlike denominators.
  4. Operator (renamed "Multiply and Divide" in the prototype): At the most advanced level, students interpret fractions as operators that, through multiplication and division, transform an original value into a new, proportionally related value. They understand the impact of multiplying or dividing by fractions less than or greater than one.

The CRG reporting prototype visually connects student performance on the i-Ready Diagnostic scale scores to these four LP levels, allowing teachers to see how a student's numerical growth corresponds to progress through these qualitative conceptual stages.

The validity of the understanding fractions learning progression (LP) is supported by empirical evidence demonstrating a strong association between LP levels and item difficulty on the i-Ready Diagnostic assessment.

Firstly, researchers examined the alignment between the chronological order of fractions-related lessons in the i-Ready Classroom Mathematics curriculum and the estimated difficulties of corresponding i-Ready Diagnostic items. They found a moderate to strong rank correlation of 0.68 between the lesson order and the LP level of assessed skills within those lessons, indicating that as students advance through grades, they are exposed to increasingly sophisticated fraction concepts aligned with higher LP levels.

Secondly, a direct association was observed between the difficulty of individual i-Ready Diagnostic items and the LP level to which they were coded. As the LP level increased (from Part-Whole to Operator), the average item difficulty also increased, with a rank correlation of 0.60. Statistical analyses (ANOVA) further confirmed that differences in mean item difficulty by LP level were significant, even after controlling for the intended grade level of the item. This empirical ordering of item difficulty provides strong evidence that the hypothesized progression of understanding fractions, as defined by the LP, is reflected in the actual performance of students on assessment items. While some overlap in item difficulty distributions exists (meaning some lower-level items can be harder than some higher-level items), the overall trend supports the LP's structure.

During pilot tests, seven teachers with prior experience with the i-Ready Diagnostic engaged in "think-aloud" sessions, interacting with the CRG reporting prototype for understanding fractions.

Teachers generally demonstrated the ability to connect i-Ready scale scores with the learning progression (LP) levels to interpret student growth meaningfully. Many teachers, particularly when in the "student view" or when viewing prior scores in the "class view," articulated how specific scale scores corresponded to particular LP levels and the types of problems a student at that level would be expected to understand. For instance, a teacher might observe a student's score and infer, "They understand fair shares, and their next progression would be to go toward number line understanding."

While some teachers initially defaulted to their existing interpretations of i-Ready growth based on numerical targets (e.g., "they made more than a year's growth"), others transitioned to more content-referenced interpretations, discussing a student's progress in terms of their evolving conceptual understanding on the LP (e.g., "they have a decent understanding of how it can be represented on a number line. They are seeing the relationships between the fractions and decimals... they're not really ready for the multiplication and division").

All teachers understood the hierarchical distinctions between the LP levels, although some questioned the "grain size" of certain levels (e.g., Level 2, "Fair Shares," felt like "a big jump" and was perceived by some as harder to teach than Level 3, "Number Line," potentially misaligning with CCSS-M standards). The written descriptions and exemplar items provided for each LP level were generally found helpful in their sense-making process.

During the pilot testing, three primary use cases emerged for how teachers could envision integrating the CRG reporting prototype into their instructional practice:

  1. Instructional Grouping: Six out of seven teachers identified the prototype's usefulness for forming student groups for targeted instructional activities related to fractions. While most preferred creating homogenous groups based on similar understanding levels, one teacher saw value in forming heterogeneous groups to promote peer learning among students with differing understandings.
  2. Parent-Teacher Communication: Three teachers believed the prototype would be valuable for communicating with both parents and students about their progress. They suggested using it in parent-teacher conferences to easily explain what test scores mean in terms of specific learning milestones. One teacher also considered using a depersonalized "class view" to share results with students themselves.
  3. Teacher Professional Learning: Five teachers saw the CRG prototype as a tool for their own professional development and deepening their content area expertise, especially for newer teachers or those less familiar with mathematics standards. They specifically highlighted the detailed LP PDF document (accessible through the prototype) as a resource for "drilling down into the standards" and enhancing their understanding of math instruction. This aligns with the CRG framework's secondary goal of contributing to teachers' professional learning and improving their attitudes toward assessment utility.

To strengthen the validity of CRG interpretations and uses, several critical aspects need to be considered:

  1. Integration with Professional Development: Teachers require initial and ongoing professional development to fully understand and utilize the learning progression framework. This support helps them shift from a "count up points" mindset to a focus on qualitative changes in student reasoning, ensuring that CRG is used for its intended formative purposes.
  2. Choosing Exemplar Items Carefully: The exemplar items used to represent LP levels must be chosen with great care. Issues arose in the pilot with the initial exemplar items for levels 2 and 3 not perfectly reflecting their theoretical difficulty, leading to confusion. A better principle is to select items that represent the most cognitively complex application of the conceptualization at that level, or to provide 3-4 items per level to better characterize the variability in item difficulty, as implemented in later prototypes. This avoids over-reliance on a single item as an instructional target.
  3. Item Design Innovations: While the i-Ready Diagnostic primarily uses selected-response items, the ideal way to ascertain a student's thinking relative to an LP level is through open-ended tasks where students show their steps or explain their reasoning. Future improvements could involve using ordered-multiple choice formats (where options map to different LP levels for partial credit) or leveraging AI for interactive chatbots that elicit more concrete evidence of student thinking. Retrofitting existing items to an LP is a start, but new item development aligned with LP design principles is crucial.
  4. Triangulation of Evidence: CRG results should serve as a starting point for inquiry, not a definitive diagnosis. Since measurement error exists and LP level explains only a portion of item difficulty variance, a student categorized at a certain level might still struggle with some concepts. Teachers should triangulate this information with evidence from other tasks (e.g., "follow-up activities" included in later prototypes) that are not subject to the same constraints as standardized assessment items, allowing for a more comprehensive understanding of student learning.

The CRG framework represents a significant step towards developing interactive score reports that provide qualitative, instructionally meaningful interpretations of student growth in large-scale assessment programs. The initial pilot tests suggest feasibility and potential, but further research is crucial before widespread implementation.

Broader implications include:

  • Enhanced Teacher Learning: CRG has the potential to deepen teachers' content area expertise and improve their attitudes towards assessments by making test results more actionable and relevant to instructional planning.
  • Improved Student Outcomes: By enabling teachers to make conceptual adjustments to their instruction based on qualitative insights into student understanding, CRG aims to facilitate improved student learning outcomes.
  • Bridging the Gap between Assessment and Instruction: CRG attempts to break from the status quo of score reporting by encouraging teachers to ask substantive questions about student learning, moving beyond mere monitoring and evaluation.

Future research directions identified for CRG include:

  • Real Data Trials: Observing how teachers interact with CRG when seeing data from their actual students, rather than hypothetical ones, to assess the impact of personal relevance.
  • Measuring Impact on Inferences and Actions: Empirically investigating the extent to which the reporting interface leads teachers to the desired instructional inferences and subsequent actions.
  • Assessing Affective and Learning Outcomes: Evaluating whether the use of CRG positively influences teacher attitudes toward assessment, their instructional practices, and ultimately, student learning outcomes.
  • Scale Invariance and Domain Specificity: Further investigating whether vertical scales built using a more flexible IRT model or those focusing exclusively on a single content domain (e.g., fractions only) maintain the generalizability and interpretability of reference distances, as well as if vertical scales designed for broad math domains are equally defensible for interpreting growth in more focused content areas like fractions.

Ultimately, while acknowledging the inherent "messiness" of learning and growth, the CRG approach prioritizes generating substantive questions for teachers, aiming to move beyond simply providing answers to foster deeper engagement with student learning.