In this talk, we'll derive the formulas for hyperbolic sine and cosine. These functions allow you to parametrize a hyperbola in a similar way that the regular sine and cosine allow you to parametrize a circle. We'll also use sculpture to show that hyperboloids are composed entirely of straight lines.

In this presentation I will explore the various ways in which a pair of students described 3-dimensional rotational symmetries while investigating symmetry groups in the context of chemistry. During a series of teaching experiments, a pair of mathematics education graduate students were asked to develop an algorithm for classifying chemically important point groups beginning with an investigation of a few ball and stick models of molecules. The student's description of rotational symmetry evolved from highly contextualized and dependent on the location of particular atoms to a much more generalized description equivalent to a conventional definition. The progress the students made through the use of each definition of rotation is framed with the Realistic Mathematics Education design heuristic of emergent models.

We discuss our investigation into income distribution in a math content course for preservice elementary teachers. We focus on a particularly powerful task which involves a reveal of the context of racial/ethnic income disparities. Findings suggest shifts in the students' views about mathematics and the role of teachers in addressing social and political issues in mathematics courses.