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Locomoting organisms often carry loads such as captured prey or young. Load-carrying effects on high-Reynolds-number flight have been studied, but the fluid dynamics of load carrying by low-Reynolds-number microorganisms has not. We studied low-Reynolds-number load carrying using unicellular choanoflagellates, which wave a flagellum to swim and create a water current transporting bacterial prey to a food-capturing collar of microvilli. A regularized Stokeslet framework was used to model the hydrodynamics of a swimming choanoflagellate with bacterial prey on its collar. Both the model and microvideography of choanoflagellates showed that swimming speed decreases as number of prey being carried increases. Flux of water into the capture zone is reduced by bacteria on the collar, which redirect the water flow and occlude parts of the collar. Feeding efficiency (prey captured per work to produce the feeding current) is decreased more by large prey, prey in the plane of flagellar beating and prey near microvillar tips than by prey in other locations. Some choanoflagellates can attach themselves to surfaces. We found that the reduction in flux due to bacterial prey on the collars of these attached thecate cells was similar to the reduction in flux for swimmers. 

Research in prospective teachers’ development of mathematical modeling knowledge for teaching is gaining momentum. The Mathematics of Doing, Understanding, Learning, and Educating for Secondary Students [MODULE(S2)]* project developed a curriculum in modeling for teacher education that includes simulations of practice, in which prospective teachers reflect on and plan a discussion around student thinking, their models, and the contextualization of their results. We present an analysis of prospective teachers’ modeling work on the decreasing area of Indigenous reservation land in the U.S., and a simulation of practice which explores different methods for finding the area of land in connection to the injustice deeply rooted in the treatment of Indigenous people. This problem explores a critical social issue and calls for explicit attention to pedagogical knowledge in structuring discussions around the contextualization of the mathematical results. 

This study examines prospective elementary teachers’ growth and development of competencies, conceptions, and perceptions of mathematical modelling in a mathematics content course for ele- mentary teachers. A series of lessons were implemented that engaged students in the modelling process through modelling tasks. The goal was to capture prospective teachers’ thoughts and per- ceptions on the meaning of mathematical modelling and their conceptions of teaching and learning modelling at the elementary and middle school level. The research questions were: (1) How do prospective teachers translate the mathematical modelling cycle into their practice of doing modelling? and (2) How do prospective teachers’ conceptions of modelling and teaching and learning modelling evolve throughout the implementation of a series of mathematical modelling lessons? Data sources included posters, modelling reports, responses to a pre- and post-intervention questionnaire, and semi-structured interview dialogue. Data analyses included mixed methods using provisional coding, open coding, and a categorical rubric. Findings from this study indicate that (1)Prospective elementary teachers translated the modelling cycle into their practice by developing their range of modelling competencies including multiple components of the modelling cycle, and (2) They developed a professionally appro- priate conception of mathematical modelling along with productive perceptions of the benefits of teaching and learning modelling.