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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. 

A new testing system for determining soil particle size distributions (PSDs) is under development. It augments existing systems generically called “SedImaging” by the addition of a water pressure transducer near the bottom of a soil sedimentation column. The original SedImaging test provides image-based PSDs for sands, while the pressure transducer will extend the PSDs into the silt range. The new test system is called the “uSed” reflecting the measurement of water pressures (u) during a SedImaging test. This paper presents the uSed theoretical equations and the results of four pilot tests on sands to verify the theory. The four specimens were coarse sand, medium-sized sand, fine sand, and a gap-graded sand. The observed pressure decay curves were qualitatively as expected and the masses of solids predicted by the uSed theory agreed with the actual specimen masses to within an error of 2%. 

This paper introduces a new paradigm that integrates the concepts of particle abrasion and breakage. Both processes can co-occur under loading as soil particles are subjected to friction as well as collisions between particles. Therefore, the significance of this integrating paradigm lies in its ability to address both abrasion and breakage in a single framework. The new paradigm is mapped out in a framework called the particle geometry space. The x-axis corresponds to the surface-area-to-volume ratio (A/V), while the y-axis represents volume (V). This space facilitates a holistic characterization of the four-particle geometry features, that is, shape (β) and size (D) as well as surface area (A) and volume (V). Three distinct paths (abrasion, breakage, and equally occurring abrasion and breakage processes), three limit lines (breakage line, sphere line, and average shape-conserving line), and five different zones are defined in the particle geometry space. Consequently, this approach enables us to systematically relate the extent of co-occurring abrasion and breakage to the particle geometry evolution. 

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.