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Symmetry is a foundational concept in inorganic chemistry, essential for understanding molecular properties and interactions. Yet, little is known about how instructors teach symmetry or what shapes their instructional and curricular choices. To investigate this, we analyzed classroom observations from fourteen inorganic chemistry instructors from various institutions, focusing on their use of student-centered practices and emphasis on symmetry content. We then conducted semi-structured interviews to explore the reasoning behind their decisions, using the Teacher-Centered Systemic Reform (TCSR) model to interpret influences from personal factors (e.g., teaching experience), teacher thinking (e.g., beliefs about teaching and learning), and contextual factors (e.g., classroom layout). Minute-by-minute analyses of teaching revealed four instructional profiles (student-centered, high-interactive, low-interactive, and instructor-centered) and four content profiles, ranging from an emphasis on symmetry fundamentals (e.g., symmetry elements and operations, point group assignment) to symmetry applications (e.g., spectroscopy, molecular orbitals, character tables). Three themes emerged: (1) instructional approaches and content emphasis vary substantially across instructors; (2) more student-centered instructors tend to focus on foundational symmetry concepts and skills, whereas more instructor-centered instructors tend to prioritize advanced applications; and (3) instructors’ beliefs and prior experiences, more than personal and contextual factors, drive instructional decisions for teaching symmetry. 

Abstract Purpose of ReviewWe review recent advances in algorithmic development and validation for modeling and control of soft robots leveraging the Koopman operator theory. Recent FindingsWe identify the following trends in recent research efforts in this area. (1) The design of lifting functions used in the data-driven approximation of the Koopman operator is critical for soft robots. (2) Robustness considerations are emphasized. Works are proposed to reduce the effect of uncertainty and noise during the process of modeling and control. (3) The Koopman operator has been embedded into different model-based control structures to drive the soft robots. SummaryBecause of their compliance and nonlinearities, modeling and control of soft robots face key challenges. To resolve these challenges, Koopman operator-based approaches have been proposed, in an effort to express the nonlinear system in a linear manner. The Koopman operator enables global linearization to reduce nonlinearities and/or serves as model constraints in model-based control algorithms for soft robots. Various implementations in soft robotic systems are illustrated and summarized in the review.