Originating from interviews with mathematics colleagues, written accounts of mathematicians engaging with mathematics, and Wes's reflections on his own mathematical work, we describe a process that we call mathematical foresight: the imagining of a resolution to a mathematical situation and a path to that resolution. In a sense, mathematical…

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…

Modern conceptions of critique express preoccupations with obtaining certainty and truth, with establishing foundations, and with linking knowledge and rationality. Modern conceptions of critique have been challenged, and critique has taken new formats. In particular critique has developed without demands of providing foundations for knowledge and…

We introduce Lesson Play as an imaginary interaction between teacher and students presented in a form of a dialogue or play. We suggest that lesson plays are a valuable professional development tool in preparing for teaching that can be juxtaposed with, or used as a replacement for, traditional lesson planning. The article begins with an…

The author has been reading Barry Mazur's "Imagining numbers," one of a number of recent general books about mathematics. This one, typically, is said in the preface to be for "people who have no training in mathematics." It does, perhaps untypically, invoke algebraic symbolism and solve quadratic and cubic equations. More…

Gary and Tall (2001) recently suggested that mathematics can be split up into "three worlds": the embodied, the proceptual and the axiomatic. They claim that objects from each of these worlds are formed, and reasoned about, in significantly different ways. During the course of this article the author considers the three world's theory…

Compared and contrasted are the concepts intuition and logic. The ideas of conceptual thought and algorithmic thought are discussed in terms of the world as a labyrinth, intuition and time, and the structure of knowledge. (KR)