Understanding rate of change requires reasoning about measurable quantities and how one quantity changes relative to another. To support this kind of reasoning, teachers should develop the ability to notice students’ quantitative and covariational reasoning. This study examines how preservice teachers (PSTs) attend to, interpret, and respond to students’ quantitative and covariational reasoning in video-based analyses of water-filling rate-of-change tasks. Drawing on relevant research, professional noticing is examined through the lenses of quantitative reasoning and covariational reasoning. Using a design-based qualitative approach, secondary PSTs participated in structured analyses of students’ problem-solving discussions related to rate of change. Data were collected across eight semi-structured sessions. This study reports qualitative analyses from two sessions (Sessions 1 and 2) that focused on rate of change. Findings show that PSTs’ initial attending shifted from perceptual task features (e.g., pouring speed, references to time) toward identification of measurable quantities, recognition of coordination between height and volume, and comparison of equal volume increments. PSTs’ interpretations progressed from recognizing secondary-variable coordination to identifying direct covariation, and their instructional responses became more targeted and content-specific. However, challenges persisted in interpreting students’ informal and visually mediated covariational reasoning. This study contributes to research on professional noticing by integrating quantitative and covariational reasoning as analytic lenses and highlighting implications for teacher preparation in calculus education.
Limbere et al. (Sat,) studied this question.
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