In CBAL® mathematics, our goal is to evaluate students' understanding of mathematical knowledge and skills in ways that effectively support teaching and learning. We are conducting research to characterize what constitutes competence in different sub-domains of mathematics and how students progress through different levels of understanding. Our characterizations take the following forms:
- An overarching framework that defines mathematical proficiency in a way that can be used for the design of assessment, instruction and professional development. This framework is called a competency model because it identifies a set of capabilities claimed to underpin competence in this discipline.
- A series of learning progressions that describes how students' competencies are thought to develop through successive stages of understanding in relation to specific areas of mathematics. In CBAL mathematics, a learning progression is defined as a description of qualitative change in a student's level of sophistication for a key concept, process, strategy, practice or habit of mind. Change may occur due to a variety of factors, including maturation and instruction, and each progression is presumed to hold for most, but not all, students. As with all scientific research, the progressions are open to empirical verification and theoretical challenge.
These resources decompose mathematical competency into several sub-domains, each of which is comprised of content-specific knowledge and processes. For example, reasoning about proportions involves one combination of competencies, while reasoning about geometry may draw on a somewhat different combination of competencies.
Importantly, we also identify key cross-cutting processes central to competence in all areas of mathematics. For example, mathematical argumentation (which ranges from the ability to provide examples and counter-examples all the way to using formal mathematical proofs), is important no matter which area of mathematics is being studied.
Determining what constitutes mathematical competence for different sub-domains and at different levels is critical for developing valid and reliable assessments. The theoretical foundation offered by the CBAL mathematics competency model and learning progressions is intended to provide a principled structure for characterizing student achievement and growth, facilitating instructional planning, supporting the professional development of teachers and ultimately improving learning.
The CBAL Math Competency Model and Provisional Learning Progressions are a work-in-progress subject to validation.
For more information, please email RDWeb@ets.org.
Learn about the CBAL Language Arts learning progressions and their relationship to the Common Core State Standards (Flash, 4:49).
Get an overview of the CBAL research program, including how the CBAL assessment prototypes are being used in the classroom (Flash, 8:02).