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  1. This descriptive study attended to the extent to which we see evidence of the presence of four practices that promote equity and access in 141 grades 3-8 mathematics lessons in the United States. We found that lessons generally showed evidence of some incorporation of the practices but often not at the highest level. Teachers in this sample engaged in social coaching at a relatively high level, across elementary and middle school classrooms. Teachers tended to do less with respect to supporting connection and engagement between student context and the math learning environment. We also found statistically significant differences between elementary and middle school lessons in positioning students as competent and supporting a nurturing environment by proactively building relationships and productive classroom culture. We offer possible interpretations and a few brief implications of these findings. 
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  2. This study examines the utilization of cognitive interviews longitudinally over a one-year period to collectively trace raters’ response processes as they interpreted and scored with observational rubrics designed to measure teaching practices that promote equity and access in elementary and middle school mathematics classrooms. We draw on four rounds of cognitive interviews (totaling 14 interviews) that involved four raters at purposeful time points spread over the year. Findings reported in this study focus on raters’ responses about one rubric, positioning students as competent. The findings point to the complexities of utilizing observational rubrics and the need to track response processes longitudinally at multiple time points during data collection in order to attend to rater calibration and the reliability and validity of resulting rubric scores. 
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