Learning progressions have been demarcated by some for science education, or only concerned with levels of sophistication in student thinking as determined by logical analyses of the discipline. We take the stance that learning progressions can be leveraged in mathematics education as a form of curriculum research that advances a linked understanding of students learning over time through careful articulation of a curricular framework and progression, instructional sequence, assessments, and levels of sophistication in student learning. Under this broadened conceptualization, we advance a methodology for developing and validating learning progressions, and advance several design considerations that can guide research concerned with engendering forms of mathematics learning, and curricular and instructional support for that learning. We advance a two-phase methodology of (a) research and development, and (b) testing and revision. Each phase involves iterative cycles of design and experimentation with the aim of developing a validated learning progression. In particular, we gathered empirical data to revise our hypothesized curricular framework and progression and to measure change in students. thinking over time as a means to validate both the effectiveness of our instructional sequence and of the assessments designed to capture learning. We use the context of early algebra to exemplify our approach to learning progressions in mathematics education with a focus on the concept of mathematical equivalence across Grades 3-5. The domain of work on research on learning over time is evolving; our work contributes a broadened role for learning progressions work in mathematics education research and practice.

This study explores how to support Turkish–American secondary school students to co-develop covariational reasoning (CR) and representational fluency (RF) in solving contextually based quadratic function tasks in an after-school community center learning setting. We conducted a teaching experiment (n = 8) at a community center. Ongoing and retrospective analyses of classroom interaction and video transcripts revealed a culturally inclusive mathematics learning environment framework with several intertwined components: co-developing CR and RF and community-based practices. This study provides evidence that students coordinate symbolic, tabular, and graphical representations, which not only deepen their understanding of how quantities change in relation to one another but also enable them to interpret and construct representations in increasingly flexible ways. This reciprocal process of co-developing CR and RF allowed students to recognize and express quantitative relationships as meaningful functional relationships, demonstrating a dynamic interplay between reasoning about change and fluency across representations. This study situates learning within culturally inclusive learning environments and acknowledges the reflexive positionality of the teacher–researcher in relation to students. We highlight how shared community-based practices can enhance mathematics teaching and learning.

Algebra as a school subject is ill defined. Students experience algebra quite differently depending on the perspective of algebra taken by authors of the textbooks from which they learn. Through a content analysis of problems (n = 63,174) in the narrative and homework sections of six high school mathematics textbook series published in the U.S., we acquired systematic and reliable information about the algebra strand (i.e., symbolic algebra and functions) of each textbook series. We introduce plots to show the density, distribution, and sequencing of content, and present analyses of data for cognitive behavior, real-world context, technology, and manipulatives. Feedback on this study from an author of each textbook series is shared, and findings are discussed in terms of students’ opportunities to learn.

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