by Daniel Chazan, University of Maryland; William Viviani, University of Maryland; Kayla White, Paint Branch High School and University of Maryland
In 2012, 100 years after Henri Poincare’s death, the magazine for the members of the Dutch Royal Mathematical Society published an “interview” with Poincare for which he “wrote” both the questions and the answers (Verhulst, 2012). When responding to a question about elegance in mathematics, Poincare makes the famous enigmatic remark attributed to him: “Mathematics is the art of giving the same names to different things” (p. 157).
In this blog post, we consider the perspectives of learners of mathematics by looking at how students may see two uses of the word tangent—with circles and in the context of derivative—as “giving the same name to different things,” but, as a negative, as impeding their understanding. We also consider the artfulness that Poincare points to and ask about artfulness in mathematics teaching; perhaps one aspect of artful teaching involves helping learners appreciate why mathematicians make the choices that they do.
Our efforts have been in the context of a technology that asks students to give examples of a mathematical object that has certain characteristics or to use examples they create to support or reject a claim about such objects.1 The teacher can then collect those multiple examples and use them to achieve their goals.