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The method of Least Square Approximation is an important topic in some linear algebra classes. Despite this, little is known about how students come to understand it, particularly in a Realistic Mathematics Education setting. Here, we report on how students used literal symbols and equations when solving a least squares problem in a travel scenario, as well as their reflections on the least squares equation in an open-ended written question. We found students used unknowns and parameters in a variety of ways. We highlight how their use of dot product equations can be helpful towards supporting their understanding of the least squares equation. 

The method of Least Square Approximation is an important topic in some linear algebra classes. Despite this, little is known about how students come to understand it, particularly in a Realistic Mathematics Education setting. Here, we report on how students used literal symbols and equations when solving a least squares problem in a travel scenario, as well as their reflections on the least squares equation in an open-ended written question. We found students used unknowns and parameters in a variety of ways. We highlight how their use of dot product equations can be helpful towards supporting their understanding of the least squares equation. 

We investigate the influence of the host matrix on the photothermally driven actuation performance of negatively photochromic, donor−acceptor Stenhouse adduct (DASA)-based polymers. Using a modular Diels−Alder “click” platform, we designed polymeric materials with varying DASA incorporation and investigated the relationships between the material composition and the resulting physical, mechanical, and photoswitching properties. We demonstrate that increasing the DASA concentration in polymer conjugates has a dramatic effect on the material’s physical and mechanical properties, such as the glass transition temperature (Tg) and elastic modulus, as well as the photoswitching properties, which are found to be highly dependent on Tg. We establish using a simple photoresponsive bilayer that actuation performance is controlled by the bilayer stiffness rather than the photochrome incorporation of DASA. Finally, we report and compare the light-induced property changes in Tg and the elastic modulus between the materials comprising the open or closed forms of DASAs. Our results demonstrate the importance of designing a material that is stiff enough to provide the mechanical strength required for actuation under load, but soft enough to reversibly switch at the operational temperature and provide key considerations for the development of application-geared photoswitchable materials.