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Contributing to research on students' multifaceted meanings for angles (e.g., angles as ray pairs, as regions, and as turns), we report on three undergraduate students' meanings for central and inscribed angles in circles. Specifically, we characterize how these meanings govern their mathematical activities when engaging in a circle geometry task, including their experienced perturbations and reconciliation of those perturbations. Our conceptual analysis reveals that some meanings are productive for students to conceive of a reflex angle in a circle and the correspondence between a central and an inscribed angle, while other meanings are limited.

The Mathematical and Theoretical Biology Institute (MTBI) is a successful summer research experience for undergraduates, with a strong record of mentoring Ph.D. graduates, particularly, underrepresented minority students. However, the MTBI program was designed for education in research, not for research in education, and the mechanisms of the students' success has never been adequately studied. This article begins the process of studying MTBI from an educational theory perspective by examining the MTBI community model from the perspective of four researchers: Piaget, Vygotsky, Bandura, and the MTBI students themselves.

Characterizing how quantities change (or vary) in tandem has been an important historical focus in mathematics that extends into the current teaching of mathematics. Thus, how students conceptualize quantities that change in tandem becomes critical to their mathematical development. In this paper, we propose two images of change: chunky and smooth. Chunky images of change are based in imagining change as occurring countable and completed amounts, whereas smooth images of change are based in imagining a continually changing experience. Using empirical examples of student work, we illustrate these images of change and their implications.

The author discusses the enormous field of study on problem solving which is an open-ended mathematical modeling research. He states the use of problem solving in the development of mathematical modelers as well as applied mathematics researchers. Also investigated is the example of research modeling in Mathematical and Theoretical Biology Institute (MTBI), a summer Research Experience for Undergraduates (REU) program in mathematical biology high school students.