Tiling a Rectangle with the Fewest Squares

AbstractWe show that a square-tiling of ap×qrectangle, wherepandqare relatively prime integers, has at least log2psquares. Ifq>pwe construct a square-tiling with less thanq/p+Clogpsquares of integer size, for some universal constantC

Vignette of Doing Mathematics: A Meta-cognitive Tour of the Production of Some Elementary Mathematics

Mathematics educators, including some mathematicians, have, in various ways, urged that the school curriculum provide opportunities for learners to have some authentic experience of doing mathematics, opportunities to experience and develop the practices, dispositions, sensibilities, habits of mind characteristic of the generation of new mathematical knowledge and understanding – questioning, exploring, representing, conjecturing, consulting the literature, making connections, seeking proofs, proving, making aesthetic judgments, etc. (Polya 1954, Cuoco et al 2005, NCTM 2000 - Standard on Reasoning and Proof). While this inclination in curricular design has a certain appeal and merit, its curricular and instructional expressions are often contrived, or superficial, or no more than caricatures of what they are meant to emulate. One likely source of the difficulty is that most mathematics educators have little or no direct experience of doing a substantial piece of original mathematics, in part because the technical demands are often too far beyond the school curriculum. Studying the history and evolution of important mathematical developments can be helpful, but provides a less immediate and direct experience

Low-dimensional Topology and Number Theory

The workshop brought together topologists and number theorists with the intent of exploring the many tantalizing connections between these areas