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Reflection is an important part of teaching and needs to be considered carefully. In this study, we examined a mathematics instructor's reflections on teaching linear algebra. The research team employed Tall's (How humans learn to think mathematically: exploring the three worlds of mathematics. Cambridge University Press, Cambridge, 2013) framework to track the instructor's movements between the three worlds of mathematical thinking. The instructor emphasized the importance of symbolic and embodied thinking and made sure the students were ready before entering the formal world. It became evident that moving the class between the worlds appropriately at certain moments was critical to students' understanding of the concept, however at times required much anticipation and careful preparation in advance as well as creating appropriate teaching resources. The instructor's dual role as a researcher and participant resulted a model of instructional decision making which was used to give further insights during his movements between Tall's worlds and afforded him unique opportunities for reflection and subsequent carefully calculated class interventions.

In this study, a description is provided for the development of two undergraduate students' geometric reasoning about the derivative of a complex-valued function with the aid of "Geometer's Sketchpad" ("GSP") during an interview sequence designed to help them characterize the derivative geometrically. Specifically, a particular "GSP" task at the end of this interview sequence aided the participants in viewing the derivative as a local property. This advancement is notable as previous participants had been largely unable to make this characterization precisely, and the participants of this study only did so while working on the final task of the interview sequence. This task required students to determine an algebraic formula given only geometric data through "GSP." Particularly, they recognized that their disks constructed in "GSP" needed to be small to accurately identify points where the function is non-differentiable, and to stay away from the "bad points" when determining differentiability.

In this study, we analysed a mathematician's teaching journals on eigenvalues and eigenvectors in a first-year linear algebra course. The research team employed Tall's ["How humans learn to think mathematically: Exploring the three worlds of mathematics." Cambridge University Press] three-world model of embodied, symbolic and formal as a framework for understanding the mathematician and teacher's pedagogical reflections as he led the class to the formal world. In order to reach the formal world, he used a sequence of tasks that emphasized embodied and symbolic, as well as formal thinking. The analysis of the journals showed that the mathematician faced challenges in leading the class towards the formal world. The study also revealed that the mathematician strived to build a concept image, that, while perhaps mirroring his own, did not resonate with the students.

Using a diagrammatic reasoning framework about inscriptions, we explored undergraduates’ reasoning about complex-valued equations. Our findings suggest that reasoning geometrically requires first reasoning algebraically about algebraic inscriptions. We found students tended to create algebraic and geometric inscriptions when their verbiage could no longer support geometric reasoning. Furthermore, they incorporated similar iconic gestures for reasoning about their geometric inscriptions, which reduced to deictic gestures as they applied their previously developed reasoning to subsequent tasks. Contrary to other research, our participants’ gestures did not taper off with future tasks. Rather, their gestures transformed as concepts were automatized. Moreover, our research suggests that gestures serve as a link between verbiage and inscriptions rather than inscriptions serving as a link between verbiage and gesture as other researchers claim. In promoting synchronicity of algebraic and geometric reasoning, teachers may want to capitalize on the fact that students tend to implement similar gestures as they reason. [ABSTRACT FROM AUTHOR]