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In college, taking algebra can prevent degree completion. One reason for this is that algebra courses in college tend to focus on procedures disconnected from meaning-making (e.g., Goldrick-Rab, 2007). It is critical to connect procedural fluency with conceptual understanding (Kilpatrick, et al., 2001). Several instruments test algebraic proficiency, however, none were designed to test a large body of algebraic conceptions and concepts. We address this gap by developing the Algebra Concept Inventory (ACI), to test college students’ conceptual understanding in algebra. A total of 402 items were developed and tested in eight waves from spring 2019 to fall 2022, administered to 18,234 students enrolled in non-arithmetic based mathematics classes at a large urban community college in the US. Data collection followed a common-item random groups equating design. Retrospective think-aloud interviews were conducted with 135 students to assess construct validity of the items. 2PL IRT models were run on all waves; 63.4% of items (253) have at least moderate, and roughly one-third have high or very high discrimination. In all waves, peak instrument values have excellent reliability ( R ≥ 0.9 ). Convergent validity was explored through the relationship between scores on the ACI and mathematics course level. Students in “mid”-level courses scored on average 0.35 SD higher than those in “low”-level courses; students in “high”-level courses scored on average 0.35 SD higher than those in “mid”-level courses, providing strong evidence of convergent validity. There was no consistent evidence of differential item functioning (DIF) related to examinee characteristics: race/ethnicity, gender, and English-language-learner status. Results suggest that algebraic conceptual understanding, conceptualized by the ACI, is measurable. The final ACI is likely to differentiate between students of various mathematical levels, without conflating characteristics such as race, gender, etc. 

This case study of one first grade student involves the analysis of three interviews that took place before, during, and after classroom teaching experiments (CTEs). The CTEs were designed to engage children in representing algebraic concepts using graphs. Using a knowledge-in-pieces perspective, our analysis focused on identifying students’ natural intuitions and ways of thinking algebraically about a functional relationship represented using graphs. Findings reveal four seeds, two of which were identified in prior studies, and how the activation and coordination of these seeds results in students' production of function graphs. 

We illustrate how concepts from systemic functional linguistics are adapted for the analysis of multimodal representations of practice used in activities where teachers and teacher educators transact meanings about practice. We focus on the transactive register used to project practice meanings to the audience of these representations. We showcase the systems called visibility (how much of the classroom experience happening is made visible to the viewer), temporality (how sequence and duration of events are represented), and theme (how semiotic resources maintain and develop themes). We apply these systems to examine the differences between two storyboards of algebra lessons that were used in a professional development context and the different kinds of reactions teachers offered to the different storyboards.