Expanding the CBAL™ Mathematics Assessments to Elementary Grades: The Development of a Competency Model and a Rational Number Learning Progression

Abstract

Prior work on the CBAL™ mathematics competency model resulted in an initial competency model for middle school grades with several learning progressions (LPs) that elaborate central ideas in the competency model and provide a basis for connecting summative and formative assessment. In the current project, we created a competency model for Grades 3–5 that is based on both the middle school competency model and the Common Core State Standards (CCSS). We also developed an LP for rational numbers based on an extensive literature review, consultations with members of the CBAL mathematics team and other related research staff at Educational Testing Service, input from an advisory panel of external experts in mathematics education and cognitive psychology, and the use of small‐scale cognitive interviews with students and teachers. Elementary mathematical understanding, specifically that of rational numbers, is viewed as fundamental and critical to developing future knowledge and skill in middle and high school mathematics and therefore essential for success in the 21st century world. The competency model and the rational number LP serve as the conceptual basis for developing and connecting summative and formative assessment as well as professional support materials for Grades 3–5. We report here on the development process of these models and future implications for task development.

Is a Computerized Adaptive Test More Motivating Than a Fixed-Item Test?

“…Both the accuracy and speed of a response can be used by the system to continually update latent trait estimates of student mastery and fluency of core mathematical concepts and procedures. The system is part of the CBAL™ (Cognitively Based Assessment of, for, and as Learning) research initiative to develop assessments that maximize the potential for positive effects on teaching and learning (Arieli-Attali & Cayton-Hodges, 2014; Bennett, 2011). As a formative assessment, the system can provide immediate feedback on the correctness of responses and overall mastery and fluency levels.…”

The Expanded Evidence-Centered Design (e-ECD) for Learning and Assessment Systems: A Framework for Incorporating Learning Goals and Processes Within Assessment Design

“…In the ECD framework, the Student/Proficiency model defines the Knowledge, Skills, and Ability (KSA) that the assessment is targeting. Although in early publications of the ECD, it is called a Student model, in recent contexts, it is called a “Proficiency model” (e.g., Feng et al., 2009a; Almond et al., 2015), or referred to as a “Competency model” (e.g., Arieli-Attali and Cayton-Hodges, 2014; Kim et al., 2016), and it can also be perceived as a “Construct map” (Wilson, 2009). A similar notion in the field of Intelligence Tutoring Systems is a “Domain model” (Quintana et al., 2000), a “Knowledge model” (Koedinger et al., 2012; Pelánek, 2017), or a “Cognitive model” (Anderson et al., 1995).…”

“…A KSA model includes variables that are the features or attributes of competence that the assessment is targeting. The number of variables and their grain size are determined by the potential use of the assessment, and it can range from 1 (e.g., the θ in college admission tests such as the GRE, SAT, and ACT) to several subskills arranged in a map or a net (e.g., a net example, see Mislevy et al., 1999; a math competency map, see Arieli-Attali and Cayton-Hodges, 2014; two versions of a game-based physics competency model, see Kim et al., 2016). These variables can be derived by conducting a cognitive task analysis of the skill by experts, analyzing the content domain, or relying on a theory of knowledge and research findings.…”

A Learning Progression for Geometrical Measurement in One, Two, and Three Dimensions

“…This model comprises domain-specific competency models (for reading writing and mathematics), summative and formative assessments, and professional development supports. Underlying these system components is a set of learning progressions (LPs) or hypothesized trajectories of acquisition of competency 18 of which have been developed in K-12 mathematics (see Arieli-Attali & Cayton-Hodges, 2014, for rational numbers; Graf, 2009, for an initial competency model and its relationship to LPs).As part of the CBAL initiative in mathematics, we developed a hypothesized LP for geometrical measurement in terms of one-, two-, and three-dimensional measurement (hereafter referred to as the geometrical measurement LP), based on a synthesis of empirical literature in this field and through expert review. The hypothesized geometrical measurement LP is aimed to represent a developmental progression of students' understanding and learning for one-, two-, and three-dimensional measurement regarding transitions along levels within a dimension and connections across the three dimensions.…”

“…This model comprises domain‐specific competency models (for reading writing and mathematics), summative and formative assessments, and professional development supports. Underlying these system components is a set of learning progressions (LPs) or hypothesized trajectories of acquisition of competency 18 of which have been developed in K–12 mathematics (see Arieli‐Attali & Cayton‐Hodges, , for rational numbers; Graf, , for an initial competency model and its relationship to LPs).…”