Current research and professional organizations call for greater emphasis on reasoning and sense making in algebra (Chazan, 2000; Cuoco, Goldenberg, & Mark, 1996; Harel & Sowder, 2005; National Council of Teachers of Mathematics [NCTM], 2009, 2010). This paper illustrates how students in an Algebra II class had opportunities to develop…

Computational estimation is an important skill necessary for students' mathematical development. Students who can estimate well for computations rely on an understanding of many mathematical topics, including a strong number sense, which facilitates understanding the mathematical operations and contextual evidence within a problem. In turn,…

Various stakeholders in mathematics education have called for increasing the role of reasoning and proving in the school mathematics curriculum. There is some evidence that these recommendations have been taken seriously by mathematics educators and textbook developers. However, if we are truly to realize this goal, we must pose problems to…

This article confronts the issue of why secondary and post-secondary students resist accepting the equality of 0.999... and 1, even after they have seen and understood logical arguments for the equality. In some sense, we might say that the equality holds by definition of 0.999..., but this definition depends upon accepting properties of the real…

In this article, I describe a classroom investigation in which a group of prospective secondary mathematics teachers discovered theorems related to the power of a point using "The Geometer's Sketchpad" (GSP). The power of a point is defines as follows: Let "P" be a fixed point coplanar with a circle. If line "PA" is a secant line that intersects…

Parallels between martial arts and mathematics are explored. Misguided public perception of both disciplines, students' misconceptions, and the similarities between proofs and katas are among the striking commonalities between martial arts and mathematics. The author also reflects on what he has learned in his martial arts training, and how this…

In this article I report on a study of the cognitive tools that research mathematicians employ when developing deep understandings of abstract mathematical definitions. I arrived at several conclusions about this process: Examples play a predominant role in understanding definitions. Equivalent reformulations of definitions enrich understanding.…

This paper reports on an experiment studying twelfth grade students' understanding of Rolle's Theorem. In particular, we study the influence of different concept images that students employ when solving reasoning tasks related to Rolle's Theorem. We argue that students' "container schema" and "motion schema" allow for rich concept images.…

Tasks regularly completed by elementary teachers reveal the mathematical nature of their work. However, preservice teachers demonstrate a lack of depth of mathematical thought. This study investigated the criteria preservice teachers intuitively used to evaluate algorithms. The intent was to use that knowledge as a foundation for modeling…

Mathematical creativity ensures the growth of mathematics as a whole. However, the source of this growth, the creativity of the mathematician, is a relatively unexplored area in mathematics and mathematics education. In order to investigate how mathematicians create mathematics, a qualitative study involving five creative mathematicians was…